Long solitary internal waves in stable stratifications W. B. Zimmerman, J. M. Rees To cite this version: W. B. Zimmerman, J. M. Rees. Long solitary internal waves in stable stratifications. Nonlinear Processes in Geophysics, European Geosciences Union (EGU), 2004, 11 (2), pp.165-180. hal-00302313 HAL Id: hal-00302313 https://hal.archives-ouvertes.fr/hal-00302313 Submitted on 14 Apr 2004 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Nonlinear Processes in Geophysics (2004) 11: 165–180 SRef-ID: 1607-7946/npg/2004-11-165 Nonlinear Processes in Geophysics © European Geosciences Union 2004 Long solitary internal waves in stable stratifications W. B. Zimmerman1 and J. M. Rees2 1Department of Chemical and Process Engineering, University of Sheffield, Sheffield S1 3JD, UK 2Department of Applied Mathematics, University of Sheffield, Sheffield S3 7RH, UK Received: 2 December 2003 – Accepted: 21 January 2004 – Published: 14 April 2004 Abstract. Observations of internal solitary waves over an 1994), we revisit the classical analysis leading to the conclu- antarctic ice shelf (Rees and Rottman, 1994) demonstrate sion that large amplitude waves can be induced in the infin- that even large amplitude disturbances have wavelengths that ity long limit. Scorer (1949) pointed out that longer waves are bounded by simple heuristic arguments following from than a particular value selected by the shear and stratification the Scorer parameter based on linear theory for wave trap- profiles, radiate vertically, and thus are not trapped by the ping. Classical weak nonlinear theories that have been ap- waveguide. This condition is the modification due to shear plied to stable stratifications all begin with perturbations of the classical Brunt-Vais¨ al¨ a¨ frequencyfor vertical radiation of simple long waves, with corrections for weak nonlin- of waves in a purely stratified fluid. In this paper, we take earity and dispersion resulting in nonlinear wave equations into account this limitation and pose a derivation of the non- (Korteweg-deVries (KdV) or Benjamin-Davis-Ono) that ad- linear evolution equation (NEE) which reduces to the clas- mit localized propagating solutions. It is shown that these sical Korteweg-de Vries equation in the case that the Scorer theories are apparently inappropriate when the Scorer pa- wavenumber approaches zero so that infinitely long waves rameter, which gives the lowest wavenumber that does not are trapped. radiate vertically, is positive. In this paper, a new nonlinear Benney (1966) and Benjamin (1966) independently de- evolution equation is derived for an arbitrary wave packet rived the first NEE for large amplitude solitary disturbances thus including one bounded below by the Scorer parameter. in a stable stratification with shear flow. They both assumed The new theory shows that solitary internal waves excited in weak nonlinearity and weak dispersion due to a long wave high Richardson number waveguides are predicted to have a approximation. The derived NEE, which applies equally to halfwidth inversely proportional to the Scorer parameter, in either temperature or streamfunction disturbances, takes the agreement with atmospheric observations. A localized an- form of a Korteweg-deVries (KdV) equation, with coeffi- alytic solution for the new wave equation is demonstrated, cients that depend on both the conditions of the waveguide and its soliton-like properties are demonstrated by numerical (functionals of the stratification and shear profiles) and on simulation. the characteristics of the disturbance itself (wavelength and amplitude): ∂A ∂A ∂3A + (c + A) + σ = 0 . (1) 1 Introduction ∂t ∂x ∂x3 Chemical engineering abounds with unit operations where This has the solution that the amplitude A, at time t and po- solitary waves can be induced. Film flows, especially in con- sition x, is densers and distillation columns, give rise to sheared, sta- 1/2 ! ble stratifications where solitary waves can be induced by 2 A0 A = 3A0sech (x − (c + A0)t) (2) mass transfer (Hewakandamby and Zimmerman, 2003) or 4σ heat transfer effects that are driven by surface tension gradi- ents. In this paper, motivated by recent studies of surfactant where A0 is the amplitude of the initial disturbance, c the 2 spreading on sheared, stable stratifications and observations phase velocity of infinitely long waves, and σ = cH /6 is the in the stable Antarctic boundary layer (Rees and Rottman, parameter measuring dispersion. H is the layer depth. The −1/2 halfwidth of the solitary wave is (A0/16√σ) . Because this Correspondence to: W. B. Zimmerman halfwidth is inversely proportional to σ, it follows that the ([email protected]) wavelength of a localised disturbance depends inversely on 166 W. B. Zimmerman and J. M. Rees: Long solitary waves the spectral width of the wave packet that characterizes the in the waveguide, it is not at first clear why large amplitude Fourier power spectrum of the disturbance. solitary disturbances exhibit the Scorer wavenumber. The KdV theory proposed by Benney and Benjamin is Ursell (1953) resolved a long standing paradox in the the- robust, with many applications in atmospheric and oceano- ory of solitary surface gravity waves by coining a new dimen- 2 3 2 graphic analysis, e.g. Doviak et al. (1991). Nevertheless, sionless number Ur = A0L /H = ε/µ , the ratio of two di- there is a fundamental problem with the derivation of the mensionless numbers. ε = A0/H is the ratio of the maximum KdV theory from the infinitely long wave limit. Essentially, streamfunction displacement to the height H of the wave- in some circumstances, the waveguide will be depleted of guide. µ = H/L is the ratio of the height of the waveguide wave energy in long scales. This is described more fully be- to the expected wavelength L of the long wave disturbance. low. Benney’s derivation begins with a linear wave equation Thus, ε and µ are properties of the resultant wave generated that to leading order is a simple wave of infinitely long wave- from some initial disturbance to the waveguide. Since the length, satisfying: formation of a solitary wave may occur after a combination of the nonlinear breaking of the initial disturbance or the dis- ∂A ∂A + c = 0 (3) persion of different components, the length scales of the ini- ∂t ∂x tial conditions may have little bearing on the resultant wave. Scorer (1949), however, considers modified Brunt-Vais¨ al¨ a¨ Thus, the description of solitary wave motion by the param- vertical buoyancy waves in the presence of shear flow. He eters µ and ε ignores the history of its formation. Ursell sug- concludes that wave energy is only trapped if it is confined gested that when Ur ∼ 1, with both ε 1 and µ 1, soli- to wavenumbers k that are greater than the Scorer parameter tary waves of permanent form can propagate. In fact, this L defined here: argument previously appeared in the pioneering papers of (Boussinesq, 1871a,b, 1872) and was understood earlier by N 2 1 d2uˆ L2 = − Lord Rayleigh (Rayleigh, 1876). It should be noted, that in 2 2 (4) uˆ − ν uˆ − ν dy defining µ and ε as above, the solitary wave is presupposed to exist. On dimensional analysis grounds, µ and ε are inde- N is the Brunt-Vais¨ al¨ a¨ frequency, ν is the phase veloc- pendent parameters. The Ursell hypothesis relating the two ˆ ity, and u (y) is the background shear profile, which is as- follows from the stronger constraint that the solitary wave sumed steady before and after the passage of the disturbance. should have permanent form. The specific functional rela- Wave energy can radiate vertically out of the waveguide as tionship that Ursell proposed was physically motivated by the Brunt-Vais¨ al¨ a¨ waves if the wavenumber of the disturbance mechanism maintaining permanent form–nonlinear steepen- is smaller than L. Scorer (1949), showed that the Fourier ing balancing wave dispersion. ˆ component of the vertical velocity wk satisfies in the linear Internal solitary waves may propagate in a waveguide of approximation: a stratified fluid layer, with and without shear. However, the 2 inclusion of shear introduces a free parameter, the Richard- d wˆ k 2 2 + L − k wˆ k = 0 . (5) son number Ri, the ratio of buoyancy to shear forces, which dy2 affects the balance of nonlinearity and dispersion in wave y is the vertical coordinate and k is the wavenumber of the pro- pagation. Dimensional analysis, along with the con- disturbance. Scorer showed that when L2 decreases with straint that the waveform be permanent, results in the func- height, energy can be trapped at low levels. More generally, tional relationship ε = f (µ, Ri). The form of this functional within in any waveguide, if L2 > k2, Eq. (5) admits oscilla- relationship should be based on accounting for the relative tory solutions that radiate vertically as Brunt-Vais¨ al¨ a¨ waves. strength of shear and stratification in maintaining the balance When L2 < k2, one solution of Eq. (5) is a bound-state, i.e. between nonlinearity and dispersion in the propagation of a energy is trapped in the waveguide and can only propagate wave of permanent form. It follows that the wavelength that within the waveguide. The other solution to Eq. (5) in this should be selected in the long time asymptotic state should case is exponential growth far away from the waveguide, be that with weakest dispersion balancing weak nonlinear- which is rejected as aphysical.