The wave-induced flow of internal gravity wavepackets with arbitrary aspect ratio

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1 This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics The wave-induced flow of internal gravity wavepackets with arbitrary aspect ratio T. S. van den Bremer a and B. R. Sutherland b a School of Engineering, University of Edinburgh, Edinburgh, UK b Depts of Physics and of Earth & Atmospheric Sciences, University of Alberta, Edmonton, Alberta, Canada (Received xx; revised xx; accepted xx) We examine the wave-induced flow of small-amplitude, quasi-monochromatic, threedimensional, Boussinesq internal gravity wavepackets in a uniformly stratified ambient. It has been known since Bretherton (969) that one-, two- and three-dimensional wave packets induce qualitatively different flows. Whereas the wave-induced mean flow for compact threedimensional wavepackets consists of a purely horizontal localized circulation that translates with and around the wavepacket known as the Bretherton flow, such a flow is prohibited for a twodimensional wavepacket of infinite spanwise extent, which instead induces a non-local internal wave response that is long compared with the streamwise-extent of the wavepacket. Through perturbation theory for quasi-monochromatic wavepackets of arbitrary aspect ratio, we derive equations that captures the Bretherton flow and the long internal waves, which have a small but non-negligible vertical velocity component. From the solutions found for a Gaussian wavepacket of arbitrary vertical and horizontal aspect ratio, we compose a regime diagram that predicts whether the induced flow is comparable to that of one-, two- or compact three-dimensional wavepackets. The predictions agree well with the results of fully nonlinear three-dimensional numerical simulations.. Introduction Internal gravity waves move vertically through a continuously stratified fluid transporting momentum and irreversibly accelerating the background flow where they break. Even before breaking, however, localized internal wavepackets induce transient flows that migrate with the group velocity of the wavepacket. In part, this is a consequence of the divergence of the momentum flux, which is zero far from the wavepacket and largest near its centre. The result is a divergent-flux induced-flow u DF, which scales as the amplitude-squared of the waves. Its horizontal component, u DF, is analogous to the pseudomomentum per mass originally derived through the principle of wave action for internal waves (Bretherton 966; Bretherton & Garrett 969; Bretherton 969; Acheson 976), later determined through the generalized Lagrangian mean formulation (Andrews & McIntyre 978a,b; Bühler & McIntyre 998) and Hamiltonian fluid dynamics (Scinocca & Shepherd 992), and, perhaps most intuitively, through momentum conservation for quasi-monochromatic wavepackets (Dosser & Sutherland 2a,b), as reviewed in part in textbooks by Bühler (29a,b) and Sutherland (2). For one-dimensional wavepackets - which are horizontally periodic (in the x-direction), spanwise infinite and vertically localized - the total wave-induced flow is just u DF î, a non-divergent unidirectional flow in the x-direction. However, the structure of the wave-induced flow changes qualitatively for two- and three-dimensional wavepackets, for which the divergent velocity field u DF necessitates a response flow u RF, so that the total wave-induced flow u DF +u RF is non-divergent. Two-dimensional (spanwise-infinite) wavepackets induce long internal waves whose vertical phase speed matches the vertical group velocity of the wavepacket (Bretherton 969; Tabaei &

2 2 T. S. van den Bremer and B. R. Sutherland Akylas 27; van den Bremer & Sutherland 24). The induced horizontal velocity field across the wavepacket itself changes sign, oriented with the horizontal group velocity along the leading flank and having opposite sign on the trailing flank. In contrast, Three-dimensional wavepackets with comparable horizontal and vertical extents induce a horizontal circulation known as the Bretherton flow (Bretherton 969; Bühler & McIntyre 23; Tabaei & Akylas 27; Bühler 29a). As for one-dimensional wavepackets the wave-induced flow is strictly horizontal and oriented in the same direction as the group velocity over the vertical extent of the wavepacket. However, the response flow sets up an oppositely oriented recirculation on the spanwise flanks of the wavepacket. For both two- and three-dimensional wavepackets, momentum transport is not necessarily localized to the wavepacket, as in the circumstance of remote-recoil by the Bretherton flow (Bühler & McIntyre 23). Bridging the gap between one-, two- and three-dimensional wavepackets, here we examine flows induced by three-dimensional wavepackets of different aspect ratios, focusing upon wavepackets with wide, but not infinite, spanwise extent. The effect of the aspect ratio of the wavepacket upon induced flows has previously been discussed by Tabaei & Akylas (27), who derived differential equations governing the induced flow of round (localized in all three dimensions), flat (wide in both horizontal directions compared to the vertical), and two dimensional (spanwise infinite) wavepackets. Here our goal is to derive explicit solutions for the flow induced by a wavepacket of any aspect ratio and thence to quantify how wide a wavepacket must be for induced long waves to dominate over induced horizontally recirculating flows and to quantify how long a wavepacket must be for the induced flow to be comparable to that for a one-dimensional wavepacket. These theoretical predictions are corroborated by numerical simulations of Gaussian wavepackets. The paper is laid out as follows. After introducing the governing equations in 2, perturbation theory for quasi-monochromatic wavepackets is applied in 3. Therein, the strictly horizontal Bretherton flow (Bretherton 969) is rederived using our decomposition into the divergent-flux induced flow and the response flow. The set of equations describing this flow is then shown to constitute part of a a set of equations that also describes induced long waves. From asymptotic limits of the solutions of these equations, we construct a regime diagram that delineating in particular for what aspect wavepacket ratios the induced flow is dominated by either the Bretherton flow or induced long waves. These predictions are compared with numerical simulations in 4, and conclusions are drawn in Governing equations Ignoring Coriolis, diffusion and viscous effects, and invoking the Boussinesq approximation, the conservation equations for momentum, internal energy and a statement of incompressibility are Du Dt = p g ρ Dρ ˆk, ρ ρ Dt = wdρ, u=, (2.a,b,c) dz where u=(u,v,w) denotes the velocity vector with components in the directions of the respective unit vectors î (x), ĵ (y) and ˆk (z), p denotes the fluctuation pressure, and gravity g acts in the negative z direction. In the Boussinesq approximation, ρ is the (constant) characteristic density. The background density is ρ(z) and, assuming a uniformly stratified ambient, the squared buoyancy frequency N 2 = (g/ρ )(dρ/dz) is real and constant. It is convenient to express the density perturbation ρ in terms of the vertical displacement field ξ through ξ ρ/ρ (z), where the prime denotes a z-derivative. Then (2.b) can be rewritten as Dξ/Dt= w. By taking the curl of the momentum equation (2.a), the pressure terms can be eliminated and we obtain an equation

3 for the baroclinic generation of vorticity ζ u: Dζ Dt = ζ u N2( ξ ξ ) î y x ĵ. (2.2) Equations (2.b) and (2.2) can be combined, by taking x and y derivatives of (2.b). Substituting the result into (2.2) after taking its temporal derivative gives [ t 2 z ( t 2 + N 2 ) y t 2 z ( t 2 + N 2 ) x t 2 y t 2 x } {{ } L ] ( ) u= [ t u ζ N 2 [ x (uξ)ĵ y (uξ)î ]] ( ) + t ζ u, } {{ } F (2.3) where the linear matrix operator L acts on the velocity vector u and the terms in the righthand side vector F are non-linear in amplitude. The symbol denotes the tensor product. We will refer to (2.3) as the forcing equation for the induced mean flow, as linear (in amplitude) solutions substituted into the right hand side will, after averaging, give rise to a forcing of an order amplitude-squared mean flow on the left hand side. The equations that follow from substitution of the linear terms on the right-hand side of (2.3) are given explicitly in Appendix A Perturbation theory Here we derive the perturbation equations and their solutions using Fourier transforms for quasi-monochromatic internal wavepackets that are localized in all three dimensions, but whose amplitude envelope has arbitrary aspect ratio. Without loss of generality, it is assumed that the wavepacket travels in the x-z plane, for which the waves with wavenumber vector k=(k x,,k z ) are contained within an amplitude envelope having extent σ x, σ y, and σ z in the horizontal alongwave, spanwise and vertical directions, respectively. The waves have peak frequency given by the dispersion relation for non-rotating Boussinesq internal waves, ω=nk x / k. Unlike studies of wave beams (Kataoka & Akylas 23, 25), for which ω is constant and the waves have infinite spatial extent in the along-beam direction (perpendicular to k), we stress here that this study focuses upon wavepackets, for which the wavenumber and frequency have finite, but small, bandwidth. Section 3. begins with the definition of the scalings used to define the quasi-monochromatic wavepacket and the presentation of its polarization relations, taking into account its finite spatial extent. Realizing that the leading-order solution of (2.3) requires both a purely horizontal flow and a response to the divergent-flux induced flow that has zero vertical vorticity, in 3.2 we go on to derive the equations for the well-known Bretherton flow (Bretherton 969; Bühler & McIntyre 998, 23; Tabaei & Akylas 27). In 3.3 we formally apply perturbation theory to examine the flows induced directly by the forcing on the right-hand side of (2.3) as a superposition of purely horizontal flows and flows with a small vertical velocity component. Inspection of the equations and their Fourier transform reveals that they simultaneously describe one-, twoand round three-dimensional wavepackets. The method of inverse Fourier transforming, taking appropriate branch-cuts to remove the singularity in the integrand, is also described here. Finally, specifically for the consideration of Gaussian wavepackets in 3.4, scalings are derived that delineate between wavepackets that are effectively one-dimensional (inducing unidirectional flows), two-dimensional (inducing long-waves) and fully three-dimensional (primarily inducing horizontally recirculating flows).

4 4 T. S. van den Bremer and B. R. Sutherland 3.. Quasi-monochromatic wavepackets We introduce the slow variables X, Y and Z describing the translation of the wavepacket at the group velocity, c g (c gx,,c gz )=N(k 2 z,, k x k z )/ k 3 : X = ε x (x c gx t), Y = ε y y, Z = ε z (z c gz t), (3.a,b,c) in which the small bandwidth parameters based on the spatial scales of variation of the amplitude envelope are defined as ε x /(k x σ x ), ε y /(k x σ y ) and ε z /(k x σ z ). The case ε y = corresponds to two-dimensional (spanwise-infinite) wavepackets. A wavepacket is said to be round if ε x ε y ε z and wide if <ε y ε x ε z. Dispersion of the group occurs on the slow time scale T, which is understood to be much slower than that which drives the wave-induced flow. As a consequence, time derivatives in (2.3) that act upon the amplitude envelope not the waves themselves, can be represented at leading-order by spatial derivatives on the envelopescale denoting the translation of the wavepacket: t ε x c gx X ε z c gz Z. (3.2) We introduce a second small parameter α = k x A, in which A is the amplitude of vertical displacements at the centre of the group. At first order in α, the vertical displacement field is given by ξ () = ξ () = Re [ A(X,Y,Z,T)e i(k xx+k z z ωt) ], (3.3) in which the superscript denotes the order in α, the subscript denotes the order in the bandwidth parameters ε x, ε y, and ε z. As ε x, ε y, and ε z can be of different orders depending on the aspect ratio of the wavepacket we later choose, we emphasize that the subscript does not play the role of a formal ordering parameter, but simply denotes the leading-order correction for the packet structure of the waves. Finally, Re[] denotes taking the real part of the enclosed expression. For the vertical displacement field, we set higher-order terms in the bandwidth parameters to be zero without loss of generality. The second column of table reports the usual polarization relationships for plane periodic internal waves and the third column gives the next-order corrections accounting for the finite size of the wavepacket, which can be obtained from solving the linearized (in α) governing equations (2.) Zeroth-order solution: Bretherton flow Ultimately because plane internal waves are exact solutions of the fully nonlinear non-diffusive equations of motion, the leading-order forcing terms on the right-hand side of (2.3) enter at order α 2 ε 3, in which ε max{ε x,ε y,ε z }. This is immediately evident from the component-wise representation (2.3) given in appendix A together with values of the fields given in terms of the amplitude envelope in table. Unless ε x and ε y are both very much smaller than ε z, the dominant terms on the left-hand side of (2.3) are those with single derivatives. Thus, from the first two rows of (2.3), the leading-order flow in this case satisfies X w (2) (X,Y,Z)= and Y w (2) (X,Y,Z)=. Hence w (2) (X,Y,Z) = is the only meaningful solution at this order: any mean flow must be purely horizontal. This is a classical result first derived by Bretherton (969), and subsequently reproduced and used in studies of long-range influence and transfers of energy to large scales by localized wavepackets (Bühler & McIntyre 998, 23; Tabaei & Akylas 27; Wagner & Young 25; Xie & Vanneste 25). By writing a zero subscript on the expression for the induced vertical velocity and, in what follows, on the horizontal velocities, our intention is to indicate that the velocity u (2) is the zeroth-order solution that specifically corresponds to motion with identically zero vertically velocity. In the section that follows we will also consider the next order induced velocity u (2) for which the vertical velocity is small but not negligible. Without explicitly using the forcing on the right-hand side of (2.3), it is possible to derive the

5 5 Field wave-scale O(α ε i ) envelope-scale O(α ε i ) vertical displacement ξ () = Ae ıϕ ξ () = velocity (x-component) u () = ı Nk z k Aeıϕ u () = Nk x [ k zε x A X + k x ε z A Z ]e ıϕ k 3 velocity (y-component) v () = v () = N k z k k x ε y A Y e ıϕ velocity (z-component) w () = ı Nk x k Aeıϕ w () = Nk z [ k zε x A X + ε z k x A Z ]e ıϕ k 3 vorticity (x-component) ζ () x = ζ() x = ı N k k x ε y A Y e ıϕ vorticity (y-component) ζ () y = N k Ae ıϕ ζ () y = ı N [k xε x A X + k z ε z A Z ]e ıϕ k vorticity (z-component) ζ () z = ζ() z = TABLE. Expressions for different linear (in α) fields, as indicated in the first column, given at the scale of the waves in the second column and with small corrections in the third column. This last column accounts for the finite but relatively large extent of the wavepacket in the x, y and z directions,as prescribed through the values of ε x, ε y and ε z, respectively (captured by ε i in the column name). Values are given in terms of the amplitude envelope A of the vertical displacement field, as found through the linearized equations for internal waves propagating in the x-z plane. It is understood that the actual fields are the real parts of the tabulated expressions and the phase is given by ϕ=k x x+k z z ωt. consequent horizontal circulation around quasi-monochromatic wavepackets by following the approach of van den Bremer & Sutherland (24), an approach which lends additional physical insight into the problem. The total induced mean flow u (2), is represented as the sum of the divergent-flux induced flow u DF and the response flow u RF : u (2) = u DF + u RF. (3.4) The divergent-flux flow is determined explicitly from the advection terms in the momentum equations ( 3.2.). The response flow is then found from conditions for incompressibility and irrotational flow ( 3.2.2) Divergent-flux induced flow Generally, the divergent-flux induced flow results from the acceleration of the flow resulting from the slow spatial variations in the advection terms of the momentum equations (2.a) written in flux-form (van den Bremer & Sutherland 24): u DF t (u () u () ), (3.5) in which the overline denotes averaging over the short spatial scales of the individual waves. At leading order in ε, the time evolution of the wavepacket may be recast in terms of spatial derivatives according to (3.2). Hence, we have c g u DF = u () u (), v DF =, c g w DF = w () u (). (3.6a,b,c) Using the polarization relationships in table, we obtain from (3.6) the divergent-flux induced flow, which is oriented in the direction of the group velocity vector c g (van den Bremer & Sutherland 24): ( u DF = u DF,, c ) gz, (3.7) c gx

6 6 T. S. van den Bremer and B. R. Sutherland in which u DF = sgn(k x ) 2 N k A 2. (3.8) The horizontal component of the divergent-flux induced flow, u DF, is the well-established induced mean flow for one-dimensional wavepackets with amplitude envelope A(Z, T) except that in (3.8) the amplitude envelope is generally a function of three dimensional space and time: A A(X,Y,Z,T). The quantity u DF is equal to the pseudomomentum per mass, which historically has been represented in terms of the wave action (Bretherton 969; Acheson 976) by E k x /ω, in which E is the mean energy per unit mass, or by an expression derived from Hamiltonian fluid mechanics (Scinocca & Shepherd 992): ξζ y, in which ζ y is the spanwise vorticity. (See also the textbooks by Bühler (29a,b) and Sutherland (2).) Response and total induced horizontal flow With u DF given by (3.7) and the requirement that the flow is purely horizontal (w (2) = ), we must have w RF = w DF. The horizontal components of the total induced flow are given by the conditions that the total induced mean flow is incompressible, so that x u (2) + y v (2) =. (3.9) Furthermore, by substituting the polarization relationships in table into the right-hand side of the final row of (2.3), given explicitly in (A 3), it is apparent that ( u (2) ) ˆk=( u DF ) ˆk. In other words, the response flow is irrotational in the horizontal, and all the vertical vorticity in the mean flow u (2) is derived from the divergent-flux induced flow, so that ( u RF,H ) ˆk=. Noting that v DF = in (3.7), the latter condition is recast in terms of the vertical vorticity of the total horizontal flow being driven by the divergent-flux induced flow: x v (2) y u (2) = y u DF. (3.) We solve (3.9) and (3.) using three-dimensional Fourier transforms with respect to the unscaled translating co-ordinates ( x,ỹ, z) = (x c gx t,y,z c gz t) to give transformed velocities in terms of the wavenumber vector(κ, λ, µ). In particular, the Fourier transform of the divergentflux induced flow given by (3.8) is û DF = sgn(k x ) 8π 3 ( 2 N k ) A( x) 2 e ı(κ x+λỹ+µ z) d xdỹd z. (3.) The algebraic equations resulting from Fourier transforming (3.9) and (3.) are solved to give the total induced flow in Fourier space: u (2) = û DF λ2 /κ 2 H κλ/κ 2, (3.2) H in which κ H =(κ 2 + λ 2 ) /2. The inverse transform gives the induced flow in real space: u (2) ( x)= û DF λ2 /κ 2 H κλ/κ 2 e ı(κ x+λỹ+µ z) dκdλdµ. (3.3) H Of course, if one can separate the µ-dependence of û DF from its κ- and λ-dependence, then the integral in µ explicitly extracts the z-dependent part of u DF, N k A( z) 2 /2, from the integrals in κ and λ. The remaining double Fourier integral is the solution of the partial differential equation given by (22) in Tabaei & Akylas (27).

7 3.3. First-order solution: long waves Here we consider the explicit forcing of the induced flow by the leading-order non-zero value of F on the right-hand side of (2.3). Being of order ε 3, terms up to this order should be retained on the left-hand side of (2.3). We proceed with the ansatz that the induced flow has much larger horizontal extent than that of the wavepacket itself. This is certainly the case for long waves that are induced by a two-dimensional (spanwise infinite) wavepacket (Bretherton 969; Tabaei & Akylas 27; van den Bremer & Sutherland 24). Explicitly, we set ε=ε z and suppose ε y ε 2. Because spanwise infinite wavepackets induce long waves with slow variations in x even if ε x and ε z are comparable, it is expected that x-derivatives in the linear operator acting upon the induced mean flow u (2) in (2.3) should be much smaller than order ε x. Thus we take the order of these derivatives to be ε 2. Using (3.2) and keeping terms in the linear operator on the left-hand side of (2.3) up to O(ε 4 ) gives [ c 2 gzε 3 ZZZ ε 4 c 2 gz ZZY + N 2 ε 2 Y c 2 gzε 3 ZZZ ε 4 c 2 gz ZZX N 2 ε 2 X c 2 gzε 4 ZZY c 2 gzε 4 ZZX ] 7 u (2) = F, (3.4) where the overline on F denotes averaging over the fast scales of waves within the wavepacket. We see that the entries in the final row, corresponding to operations acting on the vertical vorticity field, are of order ε 4. In order to have as many equations as unknowns after truncation at O(ε 3 ), we replace the final row by the incompressibility condition u (2) =, which still needs to be satisfied. The polarization relationships in table allow for explicit evaluation of F, whose leadingorder non-zero terms contain triple derivatives of X, Y and Z. As in the consideration of long waves generated by two-dimensional wavepacket (van den Bremer & Sutherland 24), we neglect terms involving horizontal spatial derivatives because the response to the forcing is long compared to the wavepacket. (This is equivalent to treating the forcing as a delta-function in the horizontal). The terms involving Z-derivatives alone appear in (A 2) coming from the product of the vertical velocity with the spanwise vorticity each occurring at order α. Thus we find that the dominant forcing at O(α 2 ε 3 ) is F= N3 kxk 2 z 2 2 k 5 z z z A( x,ỹ, z) 2 ĵ. (3.5) Combining these results, retaining terms up to O(ε 3 ), recasting the result in terms of the unscaled translating co-ordinate ( x,ỹ, z) and using (3.8), we have the following: [ c 2 gz z z z N 2 ỹ c 2 gz z z z N 2 x x ỹ z ] u (2) = [ c 2 gz z z z u DF ]. (3.6) In the expression for the forcing on the right-hand side, we have used the formula for the vertical group velocity, c gz = Nk x k z / k 3. From (3.6), it is immediately evident that in the limit of horizontally periodic waves, for which the x-derivative terms vanish, the total induced flow is u (2) = u DF î, as is well-established for one-dimensional wavepackets (Bretherton 969; Acheson 976; Bühler & McIntyre 998; Akylas & Tabaei 25; Sutherland 26; Bühler 29a,b; Sutherland 2). Although (3.6) was derived under the assumption that the wavepacket is wide, we nonetheless find that in the limit assuming zero vertical induced flow, the equations for the Bretherton flow are also reproduced by (3.6). Specifically, of the equations for the induced flow u (2) = (u (2),v (2),w (2) ), those in the first two rows may be combined to eliminate w (2). As in the consideration of horizontally periodic waves, the factors of c 2 gz z z z may be cancelled from the

8 8 T. S. van den Bremer and B. R. Sutherland terms in the resulting equation yielding the condition for vorticity arising from the divergent flux induced flow (3.). Further assuming the induced vertical velocity is zero in evaluating the equation in the last row gives (3.9). With this insight, we see we can formally decompose the induced mean flows predicted by (3.6) into its (first two) subsequent orders: u (2) = u (2) + u (2), where we note that the subscript corresponds to the order of w (2) (and not to the magnitude of the whole vector). At zeroth-order, we have already found that w (2) = and u (2) is prescribed by the purely horizontal flow (3.3) found by solving (3.9) and (3.). While these equations were derived for a round wavepacket, we have shown they also arise from (3.6) in the limit where the z-derivative term in the linear operator is negligible small. This corresponds to tall (ε z max{ε x,ε y }) wavepackets. At next order, the velocity u (2) has non-negligible vertical velocity. First, we Fourier transform (3.6), solve the resulting algebraic equations and then inverse Fourier transform. This gives the following general integral formula for the wave-induced flow: u (2) = û DF (κ,λ,µ) c 2 gzµ 4 N 2 κ 2 H c 2 gzµ 4 N 2 λ 2 N 2 κλ c 2 gzκµ 3 e ı(κ x+λỹ+µ z) dκdλdµ. (3.7) The singularity in the integrand of (3.7) corresponds to hydrostatic internal waves with wavenumber (κ,λ,µ) set so that the vertical phase speed of the induced (long internal) waves equals the vertical group velocity of the wavepacket. This is the resonant long-short wave interaction examined by Tabaei & Akylas (27). Indeed, Fourier transforming the differential equation given by (3) of their paper after neglecting rotation by setting the Coriolis parameter to zero, gives an explicit solution in the form of our equation (3.7). It is the singularity in (3.7) that results in an induced flow with non-negligible vertical velocity. In order to select only outgoing waves, we follow the approach of van den Bremer & Sutherland (24), making use of Cauchy s residue theorem to integrate around the singularity in (3.7) and reduce the expression to a formula involving only a double integral. In order to apply the correct branch-cut that selects only the outgoing waves, we switch to cylindrical co-ordinates in wavenumber space, defining θ, so that (κ,λ)=κ H (cosθ,sinθ). Equation (3.7) becomes: u (2) = 2π û DF (κ H,θ,µ) c 2 gzµ 4 N 2 κ 2 H c 2 gzµ 4 N 2 κ 2 H sin 2 (θ) N 2 κ 2 H cos(θ)sin(θ) c 2 gzµ 3 κ H cos(θ) e ı(κ H r cos(θ θ p )+µ z) κ H dκ H dθdµ, (3.8) where r x 2 + ỹ 2 and θ p tan (ỹ/ x). The singularity in (3.8) is split into two terms: G c 2 gzµ 4 N 2 κ 2 = [ H 2c gz µ 2 Nκ H + c gz µ 2 ] Nκ H c gz µ 2. (3.9) This expression in the integral can be decomposed into the sum of the Cauchy principal value and a Dirac delta function (Voisin 99): G=G PV ıπ 2c gz µ 2 δ(nκ H c gz µ 2 )sgn(µ), (3.2) where the absolute value and sign of µ ensure outward propagating waves. It is the delta function that gives rise to the radiating waves in the solution of the integral with respect to κ H in (3.8): u (2) = ıπsgn(c gz) 2π c 2 gzµ 4 cos 2 (θ) ( 2N 2 û DF c 2 gzµ 4 cos(θ)sin(θ) e ı cgz µ 2 r cos(θ θp) ) N +µ z sgn(µ)dθdµ, c 2 gzµ 5 c gz cos(θ)/n (3.2)

9 in which κ H = c gz µ 2 /N in û DF. For c gz >, the condition for outward-propagating waves requires µcos(θ θ p ), so that θ p π/2 θ θ p + π/2 for µ and θ p + π/2 < θ < θ p + 3π/2 for µ < (and vice-versa for c gz < ). Separately integrating over positive and negative µ and combining the results gives: u (2) = π N 2 θ p +π/2 θ p π/2 û DF c 2 gzµ 4 cos 2 (θ) c 2 gzµ 4 cos(θ)sin(θ) c 2 gz c gz µ 5 cos(θ)/n ( cgz µ sin 2 r cos(θ θ p ) N 9 ) +µ zsgn(c gz ) dθdµ. (3.22) Having removed the singularity, the double integral in (3.22) is readily solved by standard numerical integration techniques. Together, the sum of the Bretherton flow (3.3), for which w (2) =, and (3.22), for which w (2) is small but non-zero, predicts the velocity field induced by a wavepacket of arbitrary horizontal aspect ratio R y σ y /σ x = ε x /ε y and arbitrary vertical aspect ratio R z σ z /σ x = ε x /ε z. Which of the two flows is dominant in the horizontal plane depends upon the aspect ratio of the wavepacket. This is examined below in the specific case of a Gaussian wavepacket Dominant induced flows for different wavepacket aspect ratios Although the equations above apply to any quasi-monochromatic wavepacket, here we specifically examine the predictions for Gaussian wavepackets with amplitude envelope A = A exp[ ( x 2 /σ 2 x + ỹ 2 /σ 2 y + z 2 /σ 2 z)/2], in which A is constant. Without loss of generality, we will assume that A is real and positive, k x > and that k z <, so that the vertical group velocity is positive. Our focus is to compare the largest values of the horizontal velocity associated with the Bretherton flow (3.3) with that associated with induced long waves (3.22): the former should be dominant for relatively round wavepackets (small σ y ); the latter dominant for relatively wide wavepackets (large σ y ). From (3.8), we see that the largest magnitude of the divergent-flux induced flow is u DF = N k A 2 /2. Using (3.), the Fourier transformed divergent-flux induced flow is û DF = [N k σ x σ y σ z /(6π 3/2 )] A 2 exp[ (κ 2 σ 2 x + λ 2 σ 2 y + µ 2 σ 2 z)/4]. We substitute this into (3.3) and extract the largest value of the horizontal induced circulation of the Bretherton flow, which occurs at the centre of the wavepacket. Thus we find the scaled maximum flow, U (2) u (2) (,,)/ u DF, to be (c.f of Bühler (29a)) U (2) = 4π ˆλ 2 R 2 y ˆκ2 + ˆλ 2 exp( ˆκ2 /4 ˆλ 2 /4) d ˆκdˆλ= R y +, (3.23) in which ˆκ κσ x, ˆλ λσ y and R y σ y /σ x denotes the horizontal aspect ratio of the packet. This result is plotted as the thick solid line in figure a. The induced long waves, on the contrary, have zero horizontal flow at the centre of the wavepacket, with the maximum in the stream-wise direction occurring vertically above the centre. Defining the non-dimensional horizontal flow along the z axis to be U (2) (ẑ) u (2) (,,ẑ)/ u DF, we have from (3.22): U (2) (ẑ)=ε 2 x R y R 4 z K 2 8 π π/2 [ ] exp ˆµ2 4 ε2 xk 2 R 2 y ˆµ 4 sin 2 (θ) 4R 4 ˆµ 4 cos 2 (θ)sin(ˆµẑ)dθd ˆµ z π/2 }{{} I ( ẑ, ε x KR y /R 2 ) z, (3.24) in which ẑ= z/σ z, K k 2 x k z / k 3 and R z σ z /σ x is the vertical aspect ratio of the wavepacket.

10 T. S. van den Bremer and B. R. Sutherland 2 a) Maximum Horizontal Induced Flow /R y U (2) (Bretherton flow) U (2) 2 (long waves) max(u (2) +U (2) 2 ), ǫx = /2 Numerics for ǫ x = /2 vdb&s24 (R y ) max(u (2) ) 2 4 ǫ x = /5 ǫ x = /2 ǫ x = / ẑmax R y = σ y /σ x b) Height of Maximum Flow Above Wavepacket Centre.6 /5 /2.5 / ǫ x = / R y = σ y /σ x FIGURE. [Colour online] a) Relative maximum value of the horizontal velocity and b) its vertical location predicted as a function of the horizontal aspect ratio R y of the wavepacket for the Bretherton flow (solid line), induced waves (blue dashed lines) and their combination (thick red dashed line). Values are computed for a wavepacket with k z /k x = and ε x = ε z = /5,/2,/, as indicated (R z = ). The circles indicate the results measured from numerical simulations with ε x = ε z = /2 and R y =, and. The bold diamond indicates the solution found for the numerical simulation of a two-dimensional wavepacket by van den Bremer & Sutherland (24). Consistent with the assumption that the generated waves are long relative to the x-extent of the packet, we can ignore the contribution κ 2 σ 2 x/4=κ 2 H cos 2 (θ)σ 2 x/4 in the envelope function û DF. We begin by considering the case for which R z = : the wavepacket is round in the x-z plane, but of arbitrary width in the spanwise direction. The maximum and its relative vertical location is computed numerically and is plotted as a function of R y as the blue dashed lines in figure for three different values of ε x, as indicated. In both panels, we set k z = k x. More generally, the asymptotic behaviour of U (2) is determined first by finding the maximum, as it depends upon z, of the double integral I(ẑ, ) in (3.24), as a function of ε x KR y /R 2 z. The resulting value, denoted by I ( ), is plotted in figure 2. The double integral asymptotically

11 2 I 29.7 I.3 I ( ) ǫ x KR y /R 2 z FIGURE 2. Maximum value of the double integral I(ẑ, ) in (3.24) as a function of ε x KR y /R 2 z. Asymptotic limits are indicted by the dashed lines: for., I 29.7; for, I.3. approaches a constant value for small, and varies as for large. The non-dimensional height where the maximum occurs varies only little, increasing from z.474 for small to z.596 for large. Taking a practical approach, the crossover from the small- to large- asymptotic regimes is estimated from the intersection of the two asymptotic curves, occurring for = c.35. Combining these results with the terms in front of the double integral in (3.24), we arrive at the following asymptotic approximations for the maximum induced flow in the x-direction associated with induced long waves that have non-negligible vertical velocity: max{u (2) z } 2.95 K 2 ε 2 R y x R 4 z.726 Kε x R 2 z for c, for c. (3.25) By comparing the maximum horizontal velocity of the Bretherton flow (3.23) and the limits of and the maximum horizontal velocity associated with induced long waves (3.25), we construct the regime diagram, shown in figure 3, which illustrates whether the flow induced by the wavepacket over its extent is best described by that of one-, two- or fully three-dimensional wavepackets, as it depends upon the horizontal and vertical aspect ratios R y and R z. In order to draw boundaries to delineate what is a smooth transition between the different regimes, we make the following assumptions. A wavepacket is considered to be quasi-one-dimensional if the wave-induced flow has a magnitude within percent of the divergent-flux induced flow u DF. From (3.23), this occurs for R y.. This transitional value is plotted as the vertical long-dashed line in figure 3. A wavepacket is considered to be quasi-two-dimensional if the flow induced by long waves exceeds the Bretherton flow. An estimate of the critical vertical aspect ratio of the wavepacket R z (as a function of R y ) when these flows are comparable is given by equating (3.23) and (3.25), in which the asymptotic approximations are extrapolated outside their strict domains of validity, to = c. For R y.3, the large- limit dominates in (3.25). Otherwise the small- limit

12 2 T. S. van den Bremer and B. R. Sutherland Ry. Bretherton (horizontally recirculating) flow R z R y /2 Rz = σz/σx quasi-d quasi-2d (long waves) R z R y / R y = σ y /σ x FIGURE 3. Regime diagram indicating for what vertical and horizontal aspect ratios (R z and R y, respectively) a Gaussian wavepacket will induce a flow similar to that of a one-dimensional wavepacket (medium gray), a two-dimensional wavepacket (light gray), which induces long waves and a fully three dimensional wavepacket (white), which induces a horizontally circulation known as the Bretherton flow. The long wave transition boundary is computed using (3.26) taking k z = k x (so that K = 2 3/2 ) and ε x = /2. The cross-hatched regions indicate where the quasi-monochromatic wavepacket assumption breaks down for ε x = /2 because either ε y > /5 or ε z > /5. The closed circles indicate the values of R y and R z for the three simulations with snapshots shown in figure 4. dominates. Together we find the transition boundary is given by: { R.22[K z 2 ε 2 xr y (R y + )] /4.22[Kε x ] /2 Ry /4 for R y.3,.852[kε x (R y + )] /2.852[Kε x ] /2 Ry /2 for R y.3, (3.26) where the second similarity in two respective limits follows from taking the small and large R y limits of the (+R y )-term. Equation (3.26) is plotted as the short-dashed curve in figure 3 for the case of a wavepacket with k z = k x (K = 2 3/2 ) and ε x = /2. In particular, for large R y and this choice of parameters, the transition curve between quasi-two-dimensional and quasi-three-dimensional is given approximately by R z.3ry /2. Conversely, this transition written in terms of the horizontal aspect ratio as it depends upon the vertical aspect ratio is R y.38/(kε x ) R 2 z 77.9R 2 z. It is important to keep in mind that these results rely on the starting assumption that the wavepacket is quasi-monochromatic and sufficiently wide that lateral dispersion is a negligible effect. Hence all of ε x, ε y and ε z must be much smaller than unity. For given ε x, this puts a limitation on the aspect ratios R y = σ y /σ x = ε x /ε y and R z = σ z /σ x = ε x /ε z for which the transition boundaries can be considered reliable. In particular, the cross-hatched regions in figure 3 show the values of R y < 5ε x and R z < 5ε x for which the wavepacket, respectively, is too narrow (ε y > /5) or insufficiently tall (ε z > /5) for the case ε x = /2. The quasi-onedimensional wavepacket regime and the long-wave regime in the small R y limit lie entirely within the cross-hatched regions for the value of ε x considered here. In theory one could examine cases in which ε x is so small that transitions to flows associated with one dimensional wavepackets and two dimensional wavepackets at small R y could be reliably predicted. However, in the numerical simulations that follow, we restrict our test of theory to the examination of wavepackets with ε x = /2. This is because simulations with significantly

13 3 smaller ε x would require much larger domain sizes and consequently more memory and longer computation time. Furthermore, vertically propagating wavepackets with σ x 2kx 3λ x are unlikely to be generated by a realistic geophysical system. 4. Comparison with numerical simulations The flows induced by a Gaussian wavepacket were examined using fully nonlinear numerical simulations that solved the equations given by (2.) with the addition of damping terms to the momentum and internal energy equations for the purpose of numerical stability. Specifically, under the assumption that N 2 is constant, the code evolved in time the horizontal velocity and vertical displacement fields according to: t u = x (uu) y (vu) z (wu) x P+νD(u), t v = x (uv) y (vv) z (wv) y P+νD(v), t ξ = x (uξ) y (vξ) z (wξ)+w+κd(ξ) (4.a,b,c) in which the vertical velocity w and pressure over density P p/ρ were found from the respective diagnostic equations: and 2 P z w= x u y v, (4.2) = xx (uu) yx (vu) zx (wu) xy (uv) yy (vv) zy (wv) xz (uw) yz (vw) zz (ww) N 2 z ξ. The equations were solved in Fourier space (corresponding to x, y and z) in a triply periodic domain. The nonlinear (e.g. advective) terms in the equations were computed by transforming the fields to real space, multiplying and then transforming back to Fourier space. The diffusion operator D was prescribed to act as a Laplacian operator only upon horizontal wavenumbers greater than 2k x. The diffusivities of momentum and substance acting on these high wavenumbers were set to be ν = κ =.N/kx. 2 The fields were advanced in time using a leapfrog scheme with an Euler backstep taken every 2 steps. The initial wavepacket centred at the origin had wavenumber set so that k z = k x, and the Gaussian envelope had amplitude A =./k x, and x- and z-extent σ x = σ z = 2/k x (ε x = ε z = /2). In the three simulations presented here, the relative lateral extent was k x σ y = 2, 2 and 2, thus exploring the range between round and wide wavepackets with R y =, and, bracketing the predicted critical value of R y = 77.9, as derived below (3.26). In addition to the wavepacket itself, the Bretherton flow predicted by (3.3) was superimposed upon the initial velocity field. The predicted long wave solutions were not superimposed initially in order to make clear that three-dimensional wavepackets do indeed induce long waves if the wavepacket is sufficiently wide in the spanwise direction. In all cases, the domain was set to be much larger than the anticipated spatial extent of the induced flow over the duration of the simulation, up to time t = 2/N when the wavepacket had propagated well away from its initial position centred at the origin. The simulation resolved disturbances with wavenumbers up to 4k x and 4 k z in the x and z-directions. In the case R y =, the spanwise domain range was y L y with L y = kx (k z = k x ) and resolved by wavenumbers up to 28π/L y. In the cases R y = and, the domain was substantially larger to accommodate the generation of long waves and so the resolution was reduced in order for the numerical simulations to complete in a reasonable time frame. Explicitly, for R y = and, we set L y = 8 and 8, respectively, both domains being resolved by wavenumbers up to (4.3)

14 4 T. S. van den Bremer and B. R. Sutherland (a) k x σ y = 2: y-z slice at x = c gx t kxz u N kx k x y (b) k x σ y = 2: u N kx k x y (c) k x σ y = 2: kxz kxz u N kx k x y x-z slice at y = 2 2 k x x k x x k x x FIGURE 4. [Multimedia online, which show three-dimensional representations of the induced flows as well as their cross-sections.] The wavepacket-filtered x-component of velocity determined at time Nt = 2 from three simulation initialized with a Gaussian wavepacket of maximum amplitude A =.kx centred at the origin, having vertical wavenumber k z = k x, x- and z-extents σ x = σ z = 2kx (R z = ) and spanwise extents a) σ y = 2kx (R y = ), b) 2kx (R y = ) and c) 2kx (R y = ). The left (right) panels show a cross-section of the flow in the y-z (x-z) plane through the centre of the wavepacket. The thick black ovals are drawn around one standard deviation from the centre of the wavepacket predicted to be at (c gx t,,c gz t); these ovals appear as thin vertical lines in the right-panels of b) and c) because the extent in x is so small compared to the width of the domain. The colour bars to the lower-left in the left panels indicate the magnitude of the induced flows in the left and corresponding right panels. 32π/L y. The time resolution was./n in the case R y = and.25/n in the cases R y = and. Each run took 4 days to complete running in serial on a 2.9 GHz Intel Core i5. The flows induced by the wavepacket were assessed by filtering the Fourier-modes with horizontal wavenumbers between.5k x and.5k x corresponding to the linear waves that make up the wavepacket. The inverse transform thereby revealed the flow induced by the propagating wavepacket. The result of this analysis at the end of three simulations is shown in figure 4, which plots the x-component of the induced flow at time t = 2/N, when the wavepacket had propagated well away from its initial position at the origin and is now centered at (x,y,z) (7.7,, 7.7). Consistent with predictions, the round wavepacket with σ y = σ z = σ x (R y = R z = ) induces a horizontal circulation only (the Bretherton flow). There is no trace of the Bretherton flow that was initially superimposed on the wavepacket at the origin, as it has translated with the wavepacket at the predicted group velocity. The maximum x-component of the induced flow at this time indeed occurs at the centre of the wavepacket, with a value consistent with that predicted by theory (see figure ). The right panel of figure 4a can be compared at least qualitatively with

15 the induced flow shown by line contours in the left panel of figure 4 of Tabaei & Akylas (27). Their simulation was somewhat different: it included the effects of rotation, and the vertical wavenumber was relatively smaller in magnitude. Despite these differences, two instructive comparisons can be made. First, because Tabaei & Akylas (27) did not initialize the simulation with the predicted Bretherton flow, they observe a negative, non-translating, induced flow at the origin in addition to the positive induced flow that translates with the wavepacket. Second, because their wavepacket was round in this simulation, Tabaei & Akylas (27) also did not observe the generation of long waves. Our figure indeed confirms that the maximum induced flow associated with long waves is three orders of magnitude smaller than the Bretherton flow in the case R y = and ε x = /2. In simulations with a spanwise wider wavepacket (R y large), keeping R z fixed, the magnitude of the Bretherton flow becomes smaller and the flow associated with long waves becomes larger. At R y =, the largest flow associated with long waves is predicted to be 6 percent that of the maximum Bretherton flow. Indeed a weak wave signature is seen to be superimposed on the Bretherton flow in the snapshot cross-sections taken from a numerical simulation run with σ y = 2/k x (R y =, R z = ), as shown in figure 4b. Also as predicted, long waves eventually dominate in the simulation with σ y = 2/k x (R y = > R y). In this case the x-component of the induced flow is close to zero at the centre of the wavepacket with the maximum shifted up relative to the centre of the packet, as predicted by theory and shown in figure. The small discrepancies between the simulated and predicted value of max(u (2) ) and its vertical location are attributed to the necessarily coarse spatial and temporal resolution of the simulations Conclusions Historically, the induction of the Bretherton flow and of long waves by internal wavepackets have been treated as two distinct problems, the former resulting from a round wavepacket and the latter resulting from a flat or infinitely wide wavepacket (Bretherton 969; Bühler & McIntyre 998; Tabaei & Akylas 27). Using perturbation theory for quasi-monochromatic internal wavepackets we have rederived the equations in an approach that lends physical insight into the connection and overlap of these distinct flows, one having zero vertical velocity and the other having non-negligible vertical velocity. In re deriving the Bretherton flow, we show that it results from a superposition of a divergent-flux induced flow and a response flow that has zero vertical vorticity. We demonstrate that the equations for this flow can alternately be derived by setting the vertical velocity to zero in the equations describing long waves, a fortuitous result since the long wave equations were derived assuming the wavepacket was wide, not round. We also provide explicit integral solutions for the Bretherton flow and the three-dimensional induced long waves. Combined, these predict the mean flow induced by wavepackets of arbitrary vertical and horizontal aspect ratio and, in particular, allow us to assess at what aspect ratios induced long waves dominate over the Bretherton flow. Numerical simulations affirm the predicted transition. This work has focused upon vertically propagating waves in Boussinesq fluid with uniform stratification and without background rotation. We have intentionally examined non-hydrostatic wavepackets having k x k z, because work in progress has shown that hydrostatic wavepackets in a rotating frame induce long waves are that are evanescent, their frequency being smaller than the Coriolis frequency, f. As well as consideration of Coriolis forces, future work will revisit and extend the study of Tabaei & Akylas (27) examining anelastic and weakly nonlinear effects. The methodology presented here will also be used to examine flows induced by internal modes that are vertically bounded.

16 6 T. S. van den Bremer and B. R. Sutherland Acknowledgments The authors are grateful for the three reviewers whose constructive comments helped greatly to improve this manuscript. This research was performed in part through funding from the Natural Sciences and Engineering Research Council (NSERC) of Canada. TSvdB was partially supported by the Faculty of Science at the University of Alberta through its Visiting Fellowship program. Appendix A. Explicit forcing equation Here we explicitly give the three components of the matrix equation (2.3). From substitution of the linear (in α) fields in table (both columns), including only those terms that are non-zero, we obtain: ttz v (2) +( tt + N 2 ) y w (2) = t [ x (u ζ x )+ z (w ζ x )] +N 2 y [ x (u ξ )+ y (v ξ )+ z (w ξ )] (A ) + t [ζ x x u + ζ y y u + ζ y y u ], ttz u (2) ( tt + N 2 ) x w (2) = t [ x (u ζ y + u ζ y )+ y (v ζ y )+ z (w ζ y + w ζ y )] N 2 x [ x (u ξ )+ y (v ξ )+ z (w ξ )] (A 2) + t [ζ y y v ], tty u (2) + ttx v (2) = t [ζ x x w + ζ y y w + ζ y y w ], (A 3) where we have left out the superscripts () on the linear (in α) fields on the right-hand side for convenience. REFERENCES ACHESON, D. J. 976 On over-reflexion. J. Fluid Mech. 77, AKYLAS, T. R. & TABAEI, A. 25 Resonant self-acceleration and instability of nonlinear internal gravity wavetrains. In Frontiers of Nonlinear Physics (ed. A. Litvak), pp Institute of Applied Physics. ANDREWS, D. G. & MCINTYRE, M. E. 978a An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, ANDREWS, D. G. & MCINTYRE, M. E. 978b On wave action and its relatives. J. Fluid Mech. 89, VAN DEN BREMER, T. S. & SUTHERLAND, B. R. 24 The mean flow and long waves induced by twodimensional internal gravity wavepackets. Phys. Fluids 26, 66: 23, doi:.63/ BRETHERTON, F. P. 966 The propagation of groups of internal gravity waves in a shear flow. Quart. J. Roy. Meteorol. Soc. 92, BRETHERTON, F. P. 969 On the mean motion induced by gravity waves. J. Fluid Mech. 36 (4), BRETHERTON, F. P. & GARRETT, C. J. R. 969 Wavetrains in inhomogeneous moving media. Proc. R. Soc. London, Ser. A 32, BÜHLER, O. 29a Waves and Mean Flows. Cambridge, UK: Cambridge University Press. BÜHLER, O. 29b Waves and Mean Flows, 2nd edn. Cambridge, UK: Cambridge University Press. BÜHLER, O. & MCINTYRE, M. E. 998 On non-dissipative wave-mean interactions in the atmosphere or oceans. J. Fluid Mech. 354, BÜHLER, O. & MCINTYRE, M. E. 23 Remote recoil: A new wave-mean interaction effect. J. Fluid Mech. 492, DOSSER, H. V. & SUTHERLAND, B. R. 2a Anelastic internal wavepacket evolution and stability. J. Atmos. Sci. 68,

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