Focusing of an inertia-gravity wave packet. by a baroclinic shear flow

Transcription

1 Focusing of an inertia-gravit wave packet b a baroclinic shear flow N.R. Edwards & C. Staquet LEGI, Grenoble, France August, To appear in Dnamics of Atmospheres and Oceans Abstract We investigate the interaction of a small-amplitude internal gravit wave packet in a rotating fluid with a baroclinic shear flow. Ra equations and three-dimensional direct numerical simulations of the Boussinesq equations are solved for this purpose. The shear flow has a simple structure and depends separatel on the vertical and horiontal coordinates. We focus on the situation where the intrinsic frequenc Ω of the wave packet increases as the packet propagates into the shear flow, due to the horiontal dependence of this flow. The packet is trapped in the neighbourhood of the Ω = N surface (where N is the spatiall varing Brunt-Väisälä frequenc) and the wave-induced velocit, which first decas because of dispersion, is amplified there. The ra equations show that the packet ma undergo multiple reflections within a wave guide formed b sections of the Ω = N surface, and thus penetrate into the shear flow. Three-dimensional numerical simulations of the same problem show that most of the wave packet is actuall dissipated once trapped because both the group velocit and the horiontal scale of the packet have strongl decreased through the interaction. Consequentl, the wave packet is not able to travel across the shear flow, except when the latter vanishes locall. Present address: Climate and Environmental Phsics, Phsics Institute, Universit of Bern, Sidlerstrasse, CH- Bern, Switerland.

2 Introduction The purpose of this paper is to investigate how inertia-gravit waves, which are internal gravit waves modified b rotation, ma interact with a baroclinic shear flow, namel a horiontal flow with both a horiontal and a vertical shear. Such interactions occur genericall in geophsical flows, as soon as inertia-gravit waves propagate in a wind or a current or encounter a vortical motion such as a large-scale vortex. The basic interaction mechanism is elementar: the wave is refracted b the ambient flow. The ke point is that the interaction ma result in energ exchange between the wave and the ambient flow at the levels where the intrinsic wave frequenc reaches limiting values, so that the ambient flow is locall accelerated or decelerated. At those levels, the linear wave equation is either singular or degenerate (Jones []). Theoretical approaches to this problem are linear (e.g. Booker & Bretherton [], Ivanov & Moroov []) or weakl nonlinear (e.g. Brown & Stewartson []) and the WKB approximation is usuall emploed as a first step to deal with the complexit of the interaction (e.g. Bretherton [], Olbers [], Badulin & Shrira []). A classical interaction problem is that of a wave propagating upward into a verticall sheared horiontal wind U(); the medium is stabl stratified with constant Brunt-Väisälä frequenc N and the shear flow is stable (that is, the Richardson number Ri = N /(du/d) is greater than / everwhere). In this situation, the intrinsic wave frequenc Ω decreases (in absolute value) as the wave packet propagates upward. The intrinsic frequenc is the frequenc measured in a frame of reference attached to the ambient flow in which the wave propagates. The linear theor in the WKB approximation was used b Bretherton () [] to investigate the interaction process. This approximation was relaxed b Booker & Bretherton () [], permitting them to show that a monochromatic wave is absorbed b the current at the singular level where the wave phase speed along the wind matches the wind velocit (the intrinsic frequenc being ero at this level). This work was extended b Jones () [] to a rotating fluid. The intrinsic frequenc of the waves is now bounded below b the Coriolis frequenc f (that is f Ω N) and two singular levels arise where Ω = ±f. Jones showed that the behaviour of the waves is ver similar to that in the non-rotating case, the monochromatic wave being absorbed at the Ω = ±f levels. The work b Wurtele et al. () [] details the linear and nonlinear behaviour of a monochromatic wave as it approaches these singular levels, when the wave is continuousl emitted from a source at t = : as long as the wave has not reached a stead state, it crosses the Ω = ±f levels and its amplitude decreases exponentiall near the Ω = critical level; the wave is therefore evanescent there. levels. As time elapses, the wave reaches a stead state and is then absorbed at the singular If nonlinear effects are important, the wave breaks near the singular levels while being

3 partl reflected. All these results closel match the description made for the non-rotating case in the neighbourhood of the Ω = critical level b Booker & Bretherton [] and b e.g. Winters & D Asaro () [] using three-dimensional numerical simulations. Jones () [] also extended Booker & Bretherton s work to a potentiall unstable shear flow, where the Richardson number is smaller than / somewhere in the fluid. Jones [] thus showed that when the wave propagates upward such that the intrinsic frequenc increases (in absolute value), over-reflection is possible at the levels where Ω = ±N, that is, the reflection coefficient ma be greater than (for Ri > /, normal reflection occurs at the Ω = ±N levels, that is the reflection coefficient is equal to ). In summar, for a vertical shear flow U() in a non-rotating medium, in the linear theor, the wave is absorbed when Ω and reflected when Ω ±N (over-reflection ma occur if the shear flow is unstable to Kelvin-Helmholt instabilit). When the medium is rotating, the absorbing levels correspond to Ω = ±f. What happens when a monochromatic wave propagates into a horiontal shear flow U() in a rotating medium with uniform stratification? When propagation occurs so that Ω decreases, the wave behaviour depends upon the stabilit of the shear flow: when the shear flow is inertiall stable, the wave is alwas reflected, possibl with a transmitted component. When the shear flow is inertiall unstable, over-reflection ma occur (Öllers et al. []). Note the analog between the latter behaviour and that found b Jones () [] for an unstable vertical shear flow. When the monochromatic wave propagates so that Ω increases, it is trapped in the neighbourhood of vertical planes defined b Ω = ±N. The mean flow is decelerated b the interaction in this case i.e. the wave-induced energ locall increases; the wave ma then break in the neighbourhood of the Ω = ±N trapping planes, as shown b Staquet & Huerre () []. These situations are thus rather complex but still remain academic in the sense that the shear flow varies along one direction and the Brunt-Väisälä frequenc of the background medium is constant. The interaction of a wave with a baroclinic shear flow of the form U(, ) has been addressed b onl a few studies, some theoretical (Jones [], Olbers [], Badulin & Shrira []) and one experimental (Moulin & Flor []). The theoretical studies [], [] and [] rel upon the linear theor in the WKB approximation. Olbers thus computed the solutions for a wave field interacting with a balanced (thermal wind) state whose isopcnals have a non-ero constant slope. Olbers found that the waves ma be guided between the Ω = f and Ω = N planes (assuming Ω > ) and that a singular level of a ver special kind exists within this guide: this level is transparent for waves propagating from one side while approaching from the other side results in critical-laer absorption.

4 The general behaviour of a wave propagating in a shear flow which varies both along and is therefore not obvious when, for instance, Ω increases during the wave propagation: the wave should be trapped with respect to the horiontal dependence of the shear flow but should be reflected with respect to the vertical dependence. What is the actual behaviour? Our purpose is to investigate this question in a simple context. We rel on WKB theor to explore the parameter range and get some hints about the wave behaviour, and also perform three-dimensional nonlinear direct numerical simulations (DNS). The geophsical motivation of this work is to investigate whether wave-induced momentum transport can occur across the shear flow. In the present paper, we explore the wave behaviour as it propagates toward the current and qualitativel discuss the effect, if an, of the interaction on the current. The phsical model of the interacting wave packet and shear flow is described in the next section. This model is used as the initial condition of the Boussinesq equations, which are recalled in section along with the numerical method to solve them. The ra equations are briefl presented in section along with the numerical algorithm we emplo to solve these equations. Results are presented in section and conclusions are drawn in section. Phsical model The problem we model numericall consists of an inertia-gravit wave packet propagating toward and interacting with a baroclinic shear flow. The purpose of this section is to specif these two fields.. The baroclinic shear flow (U(,), B(,)) Let (x,, ), with directed upwards, be a Cartesian coordinate sstem in the rotating reference frame attached to the fluid container (this container is the numerical domain in the present case or the tank of a laborator experiment). The velocit field is denoted u = (u, v, w). The baroclinic shear flow consists of a velocit field along the x-direction U(, ) in thermal wind balance with a buoanc field B(, ): f U = B. () The baroclinic current is a horiontal shear laer with a vertical shear [ U = + tanh U ( s L s )] [ + β sin ( π H s )] ; () U is the velocit scale ( U is the speed when ), s is the horiontal location of the shear laer centre and L s is the vorticit thickness. This initial condition implies that a wave

5 packet propagating from a region where s travels, as increases, from a medium translating uniforml along the x direction with speed U, to a moving medium with both a vertical and a horiontal velocit shear. The parameter β represents the strength of the baroclinicit and H s is the tpical scale of the vertical shear. The shear flow becomes barotropic as β approaches ero. When β =, U and its first order partial derivatives vanish at planes = n defined b sin(π n /H s ) =, that is n /H s = / + n for an relative integer n. This case is discussed in section.. For β <, U/ vanishes at those planes because it changes sign while U and U/ take a minimum non-ero value there. Let B(, ) + b (x,,, t) denote the total buoanc field, where b (x,,, t) refers to the deviation from the balanced field B(, ). At t =, b is the buoanc field of the wave packet. We introduce the Brunt-Väisälä frequenc of the balanced state N defined as N = B/. The buoanc field B is obtained b integrating the thermal wind relation (): B(, ) = B () fu πβ H s ( π cos H s ) [ ( s + L s log cosh ( s L s ))]. () B () is the buoanc field of the rest state, assumed to be linear. We also define the Brunt-Väisälä frequenc of the rest state N b N = db d. N will be used as the frequenc scale hereafter, so that the time scale will be N. It is important for the purpose of our stud that the baroclinic shear flow is not subject to an intrinsic instabilit during the interaction with the wave. Let us examine which instabilities ma occur and whether the shear flow is stable against them. The incoming wave packet would act as the perturbation. In the horiontal plane, the shear flow is inertiall unstable if f(f U/ ) <. We chose the sign of U (< ) and f (> ) such that this condition is not satisfied; of course, this does not garantee that the instabilit will not develop. The shear flow ma also be unstable to Kelvin-Helmholt instabilit. This instabilit has a maximum growth rate for a wavelength along the x direction of order L s (e.g. Michalke []). Hence, this instabilit will grow for run but the growth rate is so slow that the wave-shear flow interaction will not be modified over the duration we consider. The same analsis can be used to ensure that, whatever the run, no Kelvin-Helmholt instabilit will appreciabl grow due to the vertical shear of the current (note that the latter instabilit will also be opposed b the stable stratification).. The inertia-gravit wave packet Let k = (k x, k, k ) refer to the main wave vector of an inertia-gravit wave packet and Ω = Ω(k) to the associated intrinsic frequenc. We recall the dispersion relation relating k and Ω for a

6 monochromatic wave in a fluid at rest (e.g. Leblond & Msak []): which can also be written as k H Ω = N k H k k H k + f k k, () = Ω f N Ω ; () = k H denotes the modulus of the horiontal wave vector k H = (k x, k ) and k = k. We assume that Ω > in the following. Let s denote the steepness of the wave, defined such that isopcnals are locall vertical for s = and overturned for s >. The steepness ma be defined as s = u max c x, () which is the ratio of the wave-induced velocit along the x direction to the phase velocit along that direction. At initial time t =, we assume that the wave energ is localied about the line = p, = p and confined within a wave packet of length L p along and height H p along. The steepness of the wave packet at t = is thus of the form { ( ) ( ) } p p s = s exp, () where s is the steepness at the packet centre. L p Finall, we recall the polariation relations for a monochromatic inertia-gravit wave of steepness s (see e.g. Gill [], Olbers []). These relations will be useful in section. below. Let (u, v, w, b ) be the components of the wave-induced velocit and buoanc fields and Φ = k x x + k + k ωt the phase of the wave. The polariation relations are: ( u Ωkx = s k H ( v Ωk = s kh I fk x kh H p + I fk ) kh e IΦ () ) e IΦ () w = s Ω k e IΦ () b = Is N k e IΦ. ()

7 Direct numerical simulations: equations and numerical method We solve the Navier-Stokes equations in the Boussinesq approximation for the deviation fields ( u,b ) from the balanced fields (U(, ), B(, )). These equations are: u t + ( u. )u + U u U + v x v t + ( u. )v + U v x w t + ( u. )w + U w x + w U b t + ( u. )b + U b B B + v + w x = p ρ x + fv + ν u () = p ρ fu + ν v () = ρ p + b + ν w () = κ b ().u =, () where ρ is a constant reference densit and p is the pressure deviation about the pressure field of the thermal balance state. A dimensional analsis can be briefl conducted to estimate the number of non dimensional parameters that govern the flow dnamics. Let U, k and Ω be the tpical velocit, length and time scales of the wave motion; U, L s and H s be the tpical velocit, horiontal length scale and vertical length scale of the shear flow. The wave motion also depends upon the parameters N, f, ν and κ. The flow dnamics therefore depend upon eight nondimensional parameters: For the wave field: F r w = U k N, the wave Froude number, Re w = U, the wave Renolds kν number, and the ratio Ω/N; For the shear flow: F r s = U NH s, the Froude number of the shear flow and the ratio L s H s ; For the wave-shear interaction: kh s ; For the medium: f N and P r = ν, the Prandtl number. κ Equations ( ) are solved in a rectangular domain. The boundar conditions are periodic along the x and directions and of free slip tpe along the direction. These boundar conditions allow for the use of Fourier decomposition of the fields so that an efficient numerical algorithm, of the pseudo-spectral tpe, can be emploed. Aliasing errors are removed using a standard truncation technique. Since we solve the equations of motion for the unbalanced components, the initial condition of the numerical code consists of the inertia-gravit wave packet onl. These equations are integrated forward in time using a third-order Adams-Bashforth scheme. We have performed several computations, which are listed in Table. The choice of the parameters was guided b the following point of view: the wave packet propagates into the current

8 in such a wa that its intrinsic frequenc increases because of the -dependence of the shear flow. This means that the wave packet should be trapped in the neighbourhood of the Ω = N surface and, at least for β =, ma break there. The purpose of our paper is to investigate the influence of the baroclinicit parameter β on the wave packet behaviour for < β. Note that if Ω were decreasing during the propagation, the packet phase lines would flatten and the packet would reflect in the neighbourhood of the shear flow, according to the linear stud of Öllers et al. () []. Ra theor: equations and numerical method. Background When a wave propagates in a moving fluid of velocit U and buoanc B, a phase can be locall defined at an time if the period (Ω ) and wavelength (k ) of the wave are much smaller that the scale of the spatial and temporal gradients of the background fields. This is the basic assumption of the WKB approximation. The dispersion relation holds locall, with Ω, the intrinsic frequenc of the wave, being simpl shifted b the Doppler effect: ω = Ω + k.u. () ω is the absolute frequenc of the wave, namel the frequenc measured in the frame of reference relative to which the fluid velocit is measured; in the present case, this absolute reference frame is the rotating frame of reference introduced above. In this approximation, changes in the wave properties are controlled b the gradients of the ambient fields U and N and are predicted along a ra (see f.i. Olbers []). A ra is the trajector of an observer moving with the absolute group velocit and is therefore defined b: dx i dt = c gi + U i, () where c g = Ω/ k is the intrinsic group velocit, c g + U is the absolute group velocit and thus d dt = t + (c g + U)., () refers to changes seen b an observer moving with the absolute group velocit. expression of the group velocit : ( ) Ω c g = = k i ( N Ω Ωk k x, N Ω Ωk k, Ω f Ωk k We recall the ). ()

9 Runs Parameters (L x, L, L ) (,, ) (k x, k, k )/π (/, /, ) (/, /, ) f. N P r U -. s L s β. H s s. p p L p H p ν F r w (t = ). Re w (t = ) F r s. Resolution xx xx xx Table : Parameters of the direct numerical simulations. L x, L and L are the sie of the numerical domain along the x, and directions respectivel. F r w(t = ) and Re w(t = ) are the initial Froude and Renolds numbers of the wave field and F r s is the (constant) Froude number of the shear flow. Nonspecified values are those of the computation in the preceeding column. The number of wavelengths in the wave packet is Np = L p.k /π =. along the direction and Np = H p.k /π = along the direction. All parameters are defined in section. The wave vector is refracted b the gradients of the ambient fields U and N according to the equations (expressed along a ra): dk i dt = Ω N U j k j, () N x i x i

10 that is, in the present case, with U = (U(, ),, ) and N(, ) dk x dt dk dt dk dt = () = Ω N N k U x, () = Ω N N k U x. () Since U and N do not var in time, the absolute frequenc of the wave packet remains constant along a ra. It follows that the intrinsic frequenc varies along a ra according to (using ()): dω dt du j = k j dt U dk j j dt, = k x (c g U + c g the second equalit makes use of the fact that dk x /dt =. U ); () Changes of the wave amplitude along a ra are inferred from the conservation of wave action. For an slowl varing background, the action A = E/Ω, where E is the wave energ, satisfies the conservation equation (Bretherton []): A t +.[(c g + U)A] =. () Equation () implies that the action contained in a small volume δv moving with the absolute group velocit is conserved, that is d(aδv ) dt =. () The form of the WKB theor we use is the approximation of geometrical optics but, for simplicit, the terminolog WKB approximation will be emploed throughout the paper.. A simple expression of wave-shear flow energ exchange A simple estimate of the transfers of energ between the wave-induced motions and the shear flow is provided b the WKB approximation, in the linear regime. If one assumes that the wave is of small amplitude relative to the shear flow, the Boussinesq equations ma be linearied about the thermal wind balance fields, leading to linear equations for the wave field. In the following, we assume that the mean flow is barotropic, that is, of the form U(). As is classical in linear stabilit analsis, the energetic term responsible for the interaction between the wave and the shear flow is < u v U/ >, where <> refers to some average (for instance over space). If we further assume that the WKB approximation is valid, namel the wave is locall plane and its properties var much faster than the gradients of the basic fields, then < u v du/d > ma be approximated b < u v > du/d. Using relations () and (), the latter term ma be expressed as < u v > du d = s k x k kh (Ω f ) du d, ()

11 which can also written as, using definition () of the group velocit and relation (): Using equation (), one finall gets < u v > du d = s Ω k k kh < u v du d > < u v > du d = du ( k x c g ). () d s k k kh d(ω ). () dt If dω /dt >, as in the present case, expression () implies that energ transfer occurs from the shear flow to the wave-induced motions.. Numerical method for solving the ra equations We represent the spatial form of the wave packet using a set of ras whose initial points lie in a given plane x = constant along the horiontal and vertical centre lines of the packet. We therefore need to initialie the wave vector and intrinsic frequenc at each of these points. A wave packet will generall contain a distribution of wavenumbers and a distribution of frequencies. If the packet is a result of an initial disturbance at a given (absolute) frequenc, that frequenc will remain constant if the properties of the medium do not var with time (which is the case we consider). With this in mind, we assume that the packet is characteried b a single absolute frequenc. This implies that the absolute frequenc is the same for all ras. We thus set the wave vector at the centre of the packet first and derive the absolute frequenc there from the Doppler relation (). Noting that k x is constant for all ras, the initial values of k or k must var across the packet as a result of the (weak) variations of U and N. At each point along the vertical centre line of the packet, k is evaluated b assuming that k has a constant value (equal to that at the packet centre). Similarl, along the horiontal centre line of the packet, k is computed b assuming that k keeps its value at the packet centre. The latter assumption has to be relaxed in a few cases where the implied value of k < if both ω and k are assumed constant. In such cases k is initialied to ero. Finall, the initial amplitude at each point is set according to (). Equations () and () are integrated numericall using the fourth order Runge-Kutta technique to find the position and wavenumber along the ras. We calculate the wave action, and hence the wave amplitude, from Eq. () b computing the change in volume of a small, initiall regular tetrahedral element which moves along each ra. Thus for each distinct initial point of the wave packet we calculate the paths of four ras which form the vertices of the volume element. In presenting our results we ignore the three extra ras which serve onl to allow us to calculate the change in action densit. Tpicall we use an integration step length h =. and the tetrahedral element initiall has length and height r =.. Results are qualitativel stable if these values are reduced as long as h < r.

12 Results The configuration and set of parameters we have chosen ensure that dω/dt > as the wave packet enters the shear flow. This result is not obvious a priori. Indeed, the inequalit dω/dt > is equivalent to U k x k (N f k ( ) H k k H s ) k kh ( tanhy )( + βsinz) πβcosz <, () L s with Z = π/h s and Y = ( s )/L s. In the present case, U <, k x >, k > and k H k so that this inequalit ma be approximated as (N f k ( Hs ) ( tanhy )( + βsinz) πβ k ) cosz >. () k L s k At the beginning of the computation, tanhy, Z π and k /k ±/ so that the inequalit becomes H s L s ± π β which is true for the range of parameters we consider. >, () The increase of the intrinsic frequenc during propagation is illustrated for run in Figure, using the WKB approximation. The baroclinicit parameter β is equal to. in this computation, impling that the trapping effect exists at all altitudes (i.e. U/, ). The two terms responsible for the change in Ω, namel k x c g U/ and k x c g U/, are displaed along a central ra (minus the latter term, which is negative, is actuall displaed for comparison). The same qualitative behaviour is observed at earl times for all ras. Figure shows that the propagation of the wave packet is indeed controlled b the -dependence of the shear flow so that, as we shall see, Ω increases. x k x c g U/, k x c g U/ time Figure : WKB predictions for run. The two terms responsible for the change in Ω are plotted versus time for a central ra. solid line: k xc g U/ ; dotted line: k xc g U/.

13 . Barotropic flow (β = ) When the baroclinicit parameter β is equal to, the shear flow is barotropic. WKB results are plotted for two such cases differing onl in the width of the shear laer, namel run (Figure a) and run (Figure b). The surface Ω = N, which is a vertical plane in this case, is displaed with a thick dashed line and a few illustrative contours of the velocit field are drawn using dotted lines. Figures a and b show that, for either value of L s, the ras eventuall steepen and approach the Ω = N surface tangentiall. The steepening ma easil be accounted for. The intrinsic group velocit is perpendicular to k and parallel to the ras in the (, ) plane (because the shear flow has no component in this plane). As the wave packet approaches the Ω = N surface, k increases because of the horiontal shear while k x and k keep their initial value (see Eqs. () to ()). Hence the intrinsic phase lines steepen and so do also the ras in the (, ) plane. As a consequence, the intrinsic frequenc along all ras increases with time (see relation ()), which results in the trapping of the ras at the vertical plane defined b Ω = N. All of the wave energ should accumulate at the trapping plane impling that, for a real wave field, breaking ma occur there. Light gre regions on Figure b indicate points where the wave steepness exceeds, which would be associated with overturning for a real wave field (this region is also present on Figure a but hardl visible). As shown b Booker & Bretherton () [] and Wurtele et al. () [] however, the WKB predictions with respect to energetics differ from those for a real wave field using linear theor. Hence the light gre region cannot be expected to give a reliable prediction of where, and even whether, overturning will occur. The ra paths indicated on Figures a and b give no indication of the speed of wave propagation. Since k increases monotonicall and Ω approaches N, however, it is clear from Eq. () that all components of the group velocit approach ero, hence the propagation speed decreases strongl along the ras. It follows that in practice molecular effects, which are not taken into account in the ra equations, ma cause a real wave to dissipate before overturning can occur. DNS results for run are displaed in Figure through contours of the fluctuating buoanc field. The behaviour just described is recovered: the wavelength along the -direction decreases during propagation, resulting in a sudden increase of Ω at about t = (figure c, further discussed below). As a consequence, the wave packet squashes onto the Ω = N trapping surface (figures b and c). The reduction of the wavelength in the direction is associated with an amplification of the fluctuating enstroph at the packet centre E Z, displaed in Figure. The packet centre is defined b the location x = (x,, ) where the fluctuating buoanc field b reaches a maximum

14 value. The enstroph is half the vorticit squared and E Z is therefore defined b E Z = ( u )(x ). () Figure shows that E Z increases to times its initial value in run. However, E Z first decreases b nearl a factor of, because of the dispersion of the wave packet. This dispersive effect is clearl manifested in the behaviour of the amplitude of the wave packet, as measured for instance b the maximum value of b (figure d): the amplitude first decreases b a factor of. Interaction with the shear flow and trapping at the Ω = N surface next raises the amplitude of b but the net increase from its initial value is b % onl, a factor too small to result in wave breaking. Moreover, as the wave packet slows down along the trapping surface, molecular dissipation competes with the refractive focusing of wave energ, leading to a further decrease in maximum wave steepness. We find that the packet is eventuall dissipated b molecular effects at the trapping plane. Note that the fluctuating buoanc field displaed in Figure contains the wave field and the deviation of the shear flow from its balanced initial state, as a result of the interaction with the wave. The fact that there is no noticeable non-wavelike component suggests that the latter deviation is ver small and that the displaed field ma be assumed to belong to the wave packet onl. This also implies that the shear flow is ver weakl affected b the interaction. Because the shear laer is ver narrow in this run, the intrinsic frequenc Ω stas constant until the packet reaches the trapping plane, then suddenl increases to a value around.n/f (Figure c). Some time is needed, however, for the upper bound N/f to be reached. This behaviour is ver likel due to the method we used to estimate Ω. We computed the intrinsic frequenc at the packet centre from the equalit Ω = ω k x U(x ). Since, on the right hand side, onl x is variable, discontinuities in Ω correspond to jumps in the location, x, of maximum b. The maximum jumps at t to a location closer to the Ω = N surface and slowl reaches that plane as the packet squashes. The intrinsic frequenc computed from the WKB approximation is also displaed in Figure c, for three ras. The general behaviour for each ra is close to that computed from DNS, namel Ω stas constant until the ra curves because of the change in background spatial gradients and increases to its upper bound N. The time at which Ω increases depends upon the ra location with respect to the trapping plane. Differences between WKB and DNS predictions in Figure c are thus principall a result of differences in definition of the packet centre. As far as the temporal evolution of Ω is concerned, we therefore conclude that ra trajectories are an appropriate model of the wave packet propagation in a barotropic shear flow.

15 .. Ω/f.. time (a) (b) (c) Figure : WKB predictions for run (a) and run (b). Trajectories of ras starting from the initial wave packet location for t, in a vertical (, ) plane. Light gre circles are plotted at each time the steepness along a given ra exceeds the value of. The thick dashed line marks the intersection of the Ω = N trapping surface with the (, ) plane and dotted lines represent contours of the shear flow velocit in this plane. (c) Temporal evolution of the intrinsic frequenc Ω of the wave packet normalied b the Coriolis frequenc f for run ; dashed line: WKB prediction for three ras of the packet; solid line: DNS result at the packet centre; dotted line: upper bound N/f.. Moderate baroclinicit (β =.), strong horiontal shear The parameter range for β being [, ], we explored the baroclinic flow behaviour for β =. and β =, the latter value leading to quiet regions in the flow. In the present section, results for a relativel narrow shear laer are presented (L s = ), for β =. (run ). The behaviour of the wave packet predicted b the WKB approximation and computed from direct numerical simulations for this run is displaed in Figures and respectivel. Figure a shows that k increases steadil as the wave propagates in the moving medium, reaching a value around times its initial value at the end of the DNS at t =. The vertical component k increases at about the same rate as k for < t < then decas to ero and becomes negative, impling that the ras become vertical then reflect. Since the baroclinic parameter β is not ero, the surface Ω = N does not intersect the (, ) plane in a straight vertical line. Instead, the surface flattens around the = n plane because U takes a minimum value there (Figure b). Therefore, the quasi-vertical ras meet the tilted Ω = N surface again in the = n region and reflect. The ras eventuall become guided about the = n plane, between two sections of the Ω = N surface. One ma wonder whether breaking will result since all of the wave energ should accumulate in this region.

16 (a). (b).... b (c) time (d) Figure : DNS results for run. (a) to (c) Constant contours of the fluctuating buoanc field b are plotted in a vertical (, ) plane at successive times: (a) t= (a few contours of the shear flow velocit field are indicated with dotted lines), (b) t=, (c) t=. The vertical solid line marks the Ω = N trapping plane. Note the change in densit scale between frames. (d) Value of the fluctuating buoanc field b at the packet centre, normalied b the value at t =, versus time. max E Z /max E Z (t=) time Figure : Enstroph at the centre of the wave packet, E Z, normalied b its initial value E Z(), for the main runs discussed in the paper. Solid line: run ; dashed line: run ; dash-dotted line: run ; dotted line: run.

17 k x, k, k time Ω/f.... time (a) (b) (c) Figure : WKB predictions for run. (a) Components of the wave vector versus time along a central ra: k x (dotted line), k (solid line) and k (dashed line). (b) Trajectories of ras starting from the initial wave packet location for t, in a vertical plane. See caption of Figure for the meaning of line tpes and smbols. (c) Intrinsic frequenc versus time for the same ra as in (a), normalied b f (dashed line); Ω/f computed from DNS at the packet centre is displaed for comparison (solid line) as well as N/f (dotted line). The numerical solution of the Boussinesq equations for run displas a similar behaviour at earl times. Constant contours of the fluctuating buoanc field are displaed in Figure for this run. As the wave packet propagates into the shear flow, the phase lines steepen and get trapped in the vicinit of the Ω = N surface (displaed with a solid line). The enstroph at the packet centre (Figure ) first decreases at the same rate as run because of dispersion. It next increases while the wave packet progresses along the Ω = N surface. Because the shear U /L s, which controls the interaction at earl times, is lower in run than in run, the maximum value reached b the enstroph is lower as well. Also, because the position of the trapping surface is closer to the wave packet at initial time when L s is larger, that maximum value is reached at an earl time for run. From t (Figure b), the trapping surface guides the now verticall moving wave packet toward the = n region. The packet keeps progressing downwards with a ver small group velocit and a ver small wavelength along the -direction (Figure c). When the packet again approaches the Ω = N surface, at t, the intrinsic frequenc increases (Figure c) and reaches its upper bound, equal to N. The deca of the wavelength along the direction makes the wave packet ver sensitive to viscous effects and it starts dissipating (Figure d). This fact, along with the ver small group velocit, explains wh reflection is not observed in the DNS. Indeed, assuming that the WKB prediction is quantitativel reliable, Figure a shows that reflection would first occur for t >, that is Nt/π >. In the DNS, the packet would have been totall dissipated b this time.

18 (a) (b) (c) (d) Figure : DNS results for run. Constant contours of the fluctuating buoanc field b are plotted in a vertical (, ) plane at successive times: (a) t=; (b) t=, (c) t=, (d) t=. On frame (a), a few contours of the velocit field are plotted with dotted lines. The surface Ω = N is displaed with a solid line on all frames. The intrinsic frequenc Ω computed from the WKB approximation along a central ra is also displaed in Figure c. The agreement with the DNS behaviour is good up to t. However, the central ra has not et reached the Ω = N surface at t = (but is ver close to it), which accounts for the discrepanc between the DNS result and the WKB prediction observed for t >.. Moderate baroclinicit (β =.), weak horiontal shear The penetration of the wave packet into the shear flow ma be promoted b reducing the horiontal shear U /L s. A consequence of this, however, is that a weaker amplification of the wave-induced velocit will occur. We have carried out two computations in which the parameter L s is increased to a value of and while U keeps the same value (equal to.). The same qualitative behaviour is observed and onl the results for the latter computation (run ) are presented. Results from the WKB approximation are displaed in Figure and those from DNS in Figure. It is shown in Figure a that the Ω = N surface is no longer closed near the = n plane, thereb creating a region where the trapping effect due to the horiontal shear is so weak that the

19 ras penetrate into the shear flow. As before, all ras are first trapped b the Ω = N surface and steepen; the meet that surface again in the neighbourhood of the = n plane, because it has become almost horiontal, and reflect. In doing so, the ras become trapped between two sections of that surface, which form a guide, leading the ras to penetrate further into the shear flow. The components of the wave vector are displaed in Figure b for a central ra: k steadil increases up to times its initial value while k oscillates about ero due to successive reflections of the ra. Constant contours of the fluctuating buoanc field computed from DNS are plotted at successive times for this run in Figure. The same behaviour as in the WKB approximation is observed again, up to the time the wave packet, which has steepened because of the trapping process, meets the Ω = N surface for the second time (panel d), in the neighbourhood of the = n plane. The horiontal shear is equal to at most ( β)u /L s. at this plane (we recall that U and U/ take a minimum value there as varies, for a constant ) while the wave shear is about. there. Hence, we expect the packet to further propagate into the shear flow in the neighbourhood of this plane because the horiontal shear of that flow is insignificant there. Cross-hatching of the densit field indicates a reflection of wave energ at this point, but the whole wave packet progresses so slowl that it is eventuall dissipated before it can propagate awa from the reflection region. Of course, this dissipation cannot be interpreted as an absorption at a critical level, since the intrinsic frequenc of the wave packet is close to N in this region. More precisel, Figure c shows that, as in run, Ω increases toward its upper bound in two steps, as the wave packet approaches and moves awa from the Ω = N surface. The WKB approximation is able to predict these two steps, ver likel because the central ra has just reached the Ω = N surface at the end of the DNS. Hence, a good agreement is observed with DNS result during the whole duration of this run when Ω is considered. As expected from the discussion in the previous section, the enstroph at the packet centre (displaed in Figure ) reaches a smaller maximum value than in runs and (because U /L s is smaller), but at an earl time (because the trapping plane is closer), after decaing at the same rate through dispersion. Figure suggests that the wave packet, if propagating upwards instead of downwards from its initial location, would reach the = n plane without progressing along the trapping surface Ω = N and being slowed down. Thus, we performed the same computation as run, with the onl difference that k was opposite signed (run, not shown). We found that the wave packet slowl penetrates into the shear flow b reflecting onto both sides of the Ω = N surface, as in the WKB prediction. Its amplitude has however decreased b a factor b this time so that the induced motions within the guiding region are insignificant.

20 k x, k, k t Ω/f, N/f.... time (a) (b) (c) Figure : WKB predictions for run. (a) Trajectories of ras starting from the initial wave packet location for t. See caption of Figure for the meaning of line tpes and smbols. (b) Components of the wave vector for a central ra: k x (dotted line), k (solid line) et k (dashed line) as a function of time. (c) Temporal evolution of the intrinsic frequenc Ω of the wave packet normalied b the Coriolis frequenc f; dashed line: WKB prediction for a central ra; solid line: DNS result at the packet centre; dotted line: upper bound N/f.. Strong baroclinicit (β = ), weak horiontal shear The previous computations have shown that the trapping of the wave packet b the horiontal shear of the balanced flow strongl slows down the progression of the packet. The packet, whose horiontal scale has decreased because of trapping, becomes subjected to molecular effects and is eventuall dissipated. We showed that even a ver weak trapping effect is enough to prevent the propagation of the wave packet. It follows that trapping has to be totall suppressed for the wave packet to penetrate into the shear flow. This situation is modelled in run : the baroclinit parameter β has been increased to a value of, impling that both the horiontal and vertical gradients of the mean flow (along with the mean flow itself) vanish at the = n planes. Also, the sign of the vertical component of the wave vector has been reversed to ensure that the wave packet propagates (upwards) toward the = n plane, without being trapped b and along the Ω = N surface. Results are displaed in Figure for the WKB predictions and DNS results are reported in Figure. The WKB theor predicts that, except for two ras, the wave packet should entirel penetrate into the shear flow. The vanishing of the gradients of the shear flow suppresses interaction with the latter flow and the amplitude of the wave packet is nearl unchanged; no gre region is indeed present on the ra trajectories. A qualitative agreement with the WKB predictions is now observed over the whole duration

21 (a). (b) (c) (d) Figure : DNS results for run. Constant contours of the fluctuating buoanc field b are plotted in a vertical (, ) plane at successive times: (a) t=; (b) t=, (c) t=, (d) t=. The surface Ω = N is displaed with a black line on all frames. of the DNS (Figure ). The wave packet is engulfed in the shear flow about the = n plane and propagates forward while being reflected onto the Ω = N surface. The absence of significant interaction with the shear flow is manifested in the temporal evolution of the enstroph at the packet centre (Figure ). Indeed, the enstroph increases b % at most as the packet propagates toward the shear flow and then steadil decreases. As opposed to all other runs, the intrinsic frequenc at the packet centre initiall decreases (Figure b). At t, when the packet first reflects onto the Ω = N surface, Ω suddenl grows to.n and then decreases, as the packet centre propagates toward the other side of the wave guide. The WKB approximation works surprinsingl well for t < since it also predicts that Ω decas along the central ra. We recall that the theor is valid when the wavelength is much smaller than the thickness of the shear flow. In run, the wavelength is along the direction (as indicated in Table ) and does not deca during propagation (as opposed to the other runs), while L s =. We are therefore led to conclude that a ratio of / meets the validit requirement

22 . Ω/f... (a) time (b) Figure : WKB predictions for run. (a) Trajectories of ras starting from the initial wave packet location for t. See caption of Figure for the meaning of line tpes and smbols. (b) Temporal evolution of the intrinsic frequenc Ω of the wave packet normalied b the Coriolis frequenc f; dashed line: WKB prediction for a central ra; solid line: DNS result at the packet centre; dotted line: upper bound N/f. of the theor. The discrepanc between DNS and WKB predictions for t > again corresponds to a change in the location of maximum b awa from the position of the central ra, associated with an amplification of the buoanc perturbation in the interference region around the reflection point. Conclusions The purpose of this paper was to investigate the interaction of an inertia-gravit wave packet with a unidirectional baroclinic shear flow U(, ), using the WKB approximation and three-dimensional direct numerical simulations. WKB predictions are remarkabl well verified b the DNS at earl times for all runs. The wave packet is trapped b the surface where the intrinsic frequenc reaches its upper bound (i.e., the local value of the Brunt-Väisälä frequenc), moves along this surface with its nearl-vertical group velocit and reflects from this surface at, or near, locations where U takes a minimum value along the vertical direction (hence U/ = at this location). Significant deviations from WKB predictions were found when the wave packet became strongl trapped. The wavelength along the direction of inhomogeneit ( and ), and hence also the group velocit, decreases during propagation. Molecular dissipation therefore has time to affect the small scales, competing with the refractive focusing of wave energ. In our example situation, the packet is eventuall dissipated close to the first reflection point. Onl when the shear flow vanishes locall along with its derivatives is the packet able to penetrate into the shear flow. But almost no interaction with the shear flow occurs in the latter case so that, except for momentum deposition

23 (a) (c) (b) (d)..... Figure : DNS results for run. Constant contours of the fluctuating buoanc field b are plotted in a vertical (, ) plane at successive times: (a) t= (a few contours of the shear flow velocit are displaed with dotted lines); (b) t=, (c) t=, (d) t=. The surface Ω = N is displaed with a solid line on all frames. through dissipative effects, the wave packet is not expected to induce an modification of the ambient flow. An important feature of the DNS results is the rapid dispersion of the wave packet. Dispersion and, to a lesser extent, dissipation counteract the focusing of wave energ and effectivel prevent overturning, and consequentl tracer mixing. In our simulations the packet is initiall defined as a single-wavenumber perturbation modulated b a Gaussian envelope, which is equivalent to specifing a distribution of wavenumbers. The precise influence of the form and length of the packet on the interaction is difficult to assess. In additional simulations, not shown here, we have found that similar packets propagating in unsheared flow and initiall onl one or two wavelengths long in one direction can elongate b dispersion b up to % in onl wave periods, an order of magnitude more than packets containing wavelengths (Edwards & Staquet []). Longer packets, which approximate more closel a single wavenumber, ma therefore exhibit less dispersion

24 but unfortunatel require considerabl more computing time. Wavenumber interactions could counteract dispersion even in short packets, allowing them to propagate with little change of form but exact solutions corresponding to such behaviour are known onl in certain special cases (Thorpe []). According to Badulin (, private communication) the earlier theoretical analsis of Erokhin & Sagdeev ([], [] papers in Russian) alread showed that the amplitude of an internal gravit wave packet strongl attenuates due to dispersion, thereb making the trapping process less dramatic than for plasma waves for instance, which disperse less because their group velocit behaves as /k α, with α <. The stud we have conducted is somewhat academic but geophsical implication can still be inferred. Thus, because the waves are unable to cross the shear flow barrier and are dissipated in its near neighbourhood, we expect weaker inertia-gravit wave activit to be found inside large-scale geophsical vortices than outside. Of course, more work is necessar to quantif this conjecture. Further work is currentl being performed to address cases where stronger interaction occurs. Acknowledgements We are grateful to S.I. Badulin and S.A. Thorpe for fruitful discussions. NRE was supported b the Service Hdrographique et Oceanographique de la Marine (SHOM/CMO) under Contracts.. and... Computations were performed on the national computer centre IDRIS. References [] Badulin SI, Shrira VI. On the irreversibilit of internal-wave dnamics due to wave trapping b mean flow inhomogeneities. Part. Local analsis. J. Fluid Mech. : [] Booker JR, Bretherton FP. The critical laer for internal gravit waves in a shear flow. J. Fluid Mech. : [] Bretherton FP. Propagation in slowl varing wave guides. Proc. Ro. Soc. London A:. [] Bretherton FP. The propagation of groups of internal gravit waves in a shear flow. Quart. J. Ro. Met. Soc. : [] Brown SN, Stewartson K. On the nonlinear reflection of a gravit wave at a critical level. Part. J.Fluid Mech.,. [] Edwards NR, Staquet C. In preparation.

25 [] Erokhin NS, Sagdeev RZ a. On the theor of anomalous focusing of internal waves in a two-dimensional nonuniform fluid. part. A stationar problem. Morsk. Gidrofi. J., -. [] Erokhin NS, Sagdeev RZ b. On the theor of anomalous focusing of internal waves in a horiontall inhomogeneous fluid. Part. Precise solution of the two-dimensional problem with respect to viscosit and non stationarit. Morsk. Gidrofi. J., -. [] Gill AE. Atmosphere-Ocean dnamics. International Geophsics Series, vol., Academic Press. [] Ivanov YA, Moroov YG. Deformation of internal gravit waves b a stream with horiontal shear. Okeanologie,, [] Jones WJ. Propagation of internal gravit waves in fluids with shear flow and rotation. J. Fluid Mech., [] Jones WJ. Reflexion and stabilit of waves in stabl stratified fluids with shear flow: a numerical stud. J. Fluid Mech., [] Jones WJ. Ra tracing for internal gravit waves. J. Geophs. Research, -, [] LeBlond PH, Msak LA. Waves in the Ocean. Elsevier Oceanograph Series, Elsevier Pub. [] Michalke A. On the inviscid instabilit of the hperbolictangent velocit profile, J. Fluid Mechanics.,, [] Moulin F, Flor JB. Experimental stud on the wave-breaking and mixing properties of a tall vortex. Submited to Dn. Atmos. Oceans [] Olbers DJ. The propagation of internal waves in a geostrophic current. J. Phs. Oceanogr. : [] Öllers MC, Kamp LPJ, Lott F, Van Velthoven PFJ, Kelder HM, Sluijter FW. Propagation properties of inertia-gravit waves through a barotropic shear laer and application to the Antarctic polar vortex. Quart. J. Ro. Met. Soc., -B,. [] Staquet C, Huerre G. On transport across a barotropic shear flow b breaking inertiagravit waves. Phs. Fluids, (),. [] Thorpe S.A.. On the dispersion of pairs of internal inertial gravit waves. J. Marine Research, (): -