JOURNALOF GEOPHYSICAL RESEARCH, VOL. 90, NO. C1, PAGES 889-894, JANUARY20, 1985 Conditions for Interface Surfacing, Upper Bounds on Extent of Ventilation, and Formation of Bottom Lenses Above Topography BENOIT CUSHMAN-ROISIN Mesoscale Air-Sea Interaction Group, TheFlorida State University, Tallahassee Conditions under which theinterface of a rotating, layered model can surface and form geostrophic fronts are investigated. It is shown howa simple one-dimensional, reduced-gravity model, despite its oversimplicity, can lead to surfacing criteria and upper bounds on the extentof the ventilationarea created by the separation of two newly formed fronts. Firsta no-surfacing theorem is established under restrictive conditions. Then, by relaxing conditions one by one, wind, coast, andtopography effects are investigated andfrontproperties derived. The present search for frontogenetical criteria andventilation upper bounds, which per se is a novel approach to frontal studies, isaimed at complementing the existing frontal stability studies by offering additional information onfront properties. Extension to multilayered and two-dimensional models is briefly anticipated, calling for additional work before moregeneral criteria can be formulated. 1. INTRODUCTION Reduced-gravity models continue to play an importantrole inphysical oceanography as a practica] too] for the studyof near-surface dynamics, ranging from small-scale (coastal plumes, Garvine [1984]),to mesoscale (isolated eddies, Nof [1983]), and to large-scale problems (Indian Ocean,Luther and O'Brien [1984]). Despite their restrictedvertical repre- sentation, these models have the advantage of combiningade- quate horizontal description with simple equations. Quite recently, reduced-gravity models have beenappliedto studies of geostrophic fronts,for which thef plane dynamics have a lengthscaleon the order of the radius of deformation {typically 10-50 km). In such studies the interface at the foot ofthe activelayer is allowed to surface at one or more lo- cations. At eachsurfacing location the horizontal density dis- continuity represents a front. Stern [1980'1, and later Stern et al. [1982], showed that the coastal front formingthe offshore "limit of a coastal intrusion can undergo a varietyof regimes, such as dispersion, wave steepening, and blocking.On the other hand the open-ocean front wherethe interface extends from thefront to infinity,a modelof the Gulf Stream and Kuroshio fronts, is stable to all small-amplitude perturbations [Pddor, 1983a]. The double-front structure (band of light water) isunstable to small-amplitude, longwave perturbations [Grijfiths etal.,1982], whereas Paldor [1983b] concluded that .the single-front along a coast is stable to the same pertur- hations, arguing thatthedouble-front instability is caused by monanee. While thesestudies mentioned above rely on the hypothesis ofuniform potential vorticity, Killworth and Stern [t982] demonstrated that the single front in the proximity ofa mast is unstable to infinitesimal-amplitude perturbations when the potential vorticity increases toward the coast. All these studies demonstrate that the reduced-gravity model is an adequate tool forfrontal problems. It simplifies the dynamics to aminimum, while it still allows formany frontal behaviors of practical concern. Beside studies of theevolution of one or several existing Copyright 1985 by the American Geophysical Union. Paper number 4C0989. 0148.0227/85/004C.0989505.00 fronts, the problem of frontal formation remains of interest and requires further investigation. In reduced-gravity models, frontscan appearspontaneously as the interfacesurfaces and disappear when two pools of light water collide.Oceanic ex- amples include the formation of a coastal upwelling front; the question of surfacing under intense, interfacial, internal wave activity;and the eddyshedding and reabsorption in the vicin- ity of a front-current structuresuch as the Gulf Stream. The present work is intendedto lay a new framework adapted to such investigationsand to answer partially some of the questions related to frontal appearances and resappearances. It is thus a complement to existingevolution and stability studies. As a first step a one-dimensional framework (cross-frontal coordinateand time only) is retained. The approach focuses on the surfacing problems and on the propertiesof the newly formed fronts as a result from their origin. First, the question of surfacing far away from a coast and in the absenceof forcing is examined. The answer shows that, under quite gen- eral conditions, such surfacing cannotoccur.Then, by relaxing theseconditions one by one, various surfacingproblemsare investigated, and ventilationconsiderations are developed. In particular an upper bound on the extent of ventilation (con- tact of the bottom layer with the atmosphere) is derived in some cases. 2. No-SURFACING THEOREM In this section a theorem establishes that, in the absenceof boundaries, of forcing, and of prerequisite discontinuities, sur- facing cannot occur spontaneously in the one-dimensional context. Beyond standing as a warning,this theoremis useful in determiningunder which conditionsfront formation can occur. To explore such possibilities, conditions will be relaxed one by one, and a series of frontal formation problems will follow. In the time-dependent study of a one-dimensional reduced- gravity system, one may imagine that, due to remotely forced, internalgravity wave activity on the interface, the latter may riseand eventually hit the surface, thusforming a pair of new fronts that subsequently separate. The question addressed by the present theorem is whether suchspontaneous surfacing is possible. The answer is best obtained by elementaryLa- grangian considerations. 889