IUTAM Symposium on Advances in Mathematical Modelling of Atmosphere and Ocean Dynamics

Many atmospheric and oceanic phenomena occur in a localized region. When numerically simulating such phenomena it is not practical to include all of the surrounding fluid in the numerical domain. As a case in point, one would not simulate an isolated thunderstorm with a global atmospheric model just to avoid possible problems at the lateral boundaries of a limited domain. Moreover, in a fluid such as the atmosphere there is no distinct upper boundary, and most numerical representations of the atmosphere’s vertical structure terminate at some arbitrary level.

Gent and McWilliams(1990) proposed a parameterization for the effects of mesoscale eddies on the large-scale flow for use in ocean climate models. It proposes that eddies advect tracers in addition to mixing them along isopycnal surfaces. This parameterization can be considered as a generalization of the residual-mean meridional circulation of (1976) to three dimensions. The resulting Eliassen-Palm fluxes have to be parameterized in order to determine the momentum equation used in non-eddy-resolving ocean climate models. The Antarctic Circumpolar Current region of the ocean is most like the atmosphere in that there are no continents to block the zonal jet. The nonacceleration theo rem would suggest that the eddy-induced advection of tracers should oppose mean flow advection at the latitude of Drake Passage, and this is confirmed in global ocean model simulations.

In this paper we establish the mathematical framework for the filtering problem on a bounded interval, and provide a motivation for the use of the “half-sinc” function as a boundary filter. A boundary filter is one which yields output valid at the start or end of the interval on which the input is available. Such filters have obvious relevance to the problem of initialization for Numerical Weather Prediction. We also discuss some fundamental difficulties encountered when filtering time-limited functions.

Practically our entire understanding of large-scale atmosphere-ocean dynamics depends on the notions of balance and potential-vorticity inversion. These are essential, for instance, for a clear understanding of the basic Rossby-wave propagation mechanism, or quasi-elasticity, that underlies almost every large-scale fluid-dynamical phenomenon of meteorological and oceanographical interest, from the global-scale transport of terrestrial greenhouse gases (and similar problems in the solar interior) to Rossby-wave-mediated global teleconnection, baroclinic and barotropic shear instability, vortex coherence, and vortex-core isolation. The ideas involved in understanding balance and inversion continue to hold special fascination because of their central importance both for theory and for applications, such as data assimilation, and the fact that complete mathematical understanding is still elusive. The importance for applications was adumbrated by Richardson in his pioneering study of numerical weather prediction. The importance for theory — and the exquisite subtlety involved — was adumbrated by Poincaré in his discovery of the homoclinic tangle, and by Lighthill in his discovery of the quadrupole nature of acoustic radiation by unsteady vortical motion.

The fluid mechanics of the large-scale ocean circulation when planetary scale islands and barriers are present introduces new features to the problem of the ocean circulation. Islands introduce non simply-connected domains for the ocean circulation while barriers, representing either large island arcs or mid-ocean ridges would seem to impose strong limits on the communication between oceanic sub-basins. It is shown that the application of Kelvin’s circulation theorem is an illuminating instrument in understanding both the steady and unsteady flows in the presence of such geometry. The constraint imposed by a proper application of Kelvin’s theorem leads to non intuitive predictions for the circulation around such large islands and the production of strong zonal jets in the ocean circulation. A similar application of Kelvin’s theorem to the problem of the propagation of large-scale Rossby waves yields, for example, the surprising result that barriers with just two small gaps become essentially transparent to the passage of large-scale wave energy.

To meet some of the outstanding issues associated with climate modeling such as exploiting new methodology for computation, utilizing the latest computing hardware, increasing the speed of computations and incorporating various space scales into one comprehensive model, we have developed a new global climate model entitled a Spectral Element Atmospheric Model, (SEAM), which can perform regional integrations concurrently with the global scale in a very straightforward manner. The model offers a number of distinct advantages over other global/regional models and is highly flexible. It utilizes the geometric properties of finite element methods, it allows for very convenient local mesh refinement and regional detail, it is ideally suited to parallel processing by minimizing communication amongst the processors, it is very efficient computationally, and last but not least, it has no pole problems. A number of tests with this model are described which will demonstrate the efficacy of the model’s advantageous features.

Vortices are a main feature of oceanic circulation, and thus have often been studied, mostly with a quasi-geostrophic model: stationary solutions such as tripoles have been found (Carton and Legras, 1994; Corréard and Carton, 1999). Recently, vortex stability has been investigated in a shallow-water framework (Dewar and Killworth, 1995; Carton and Baey, 2000; Stegner and Dritschel, 2000). These studies showed that the potential vorticity profile of the vortex, as well as its size (Burger number, see Benilov et al., 1998) and its intensity (Rossby number) are crucial for its stability. But they defined the vortex via its velocity profile, thus changing potential vorticity with stratification. In the present study, the potential vorticity profile of the vortex is given independently of the other parameters and is reduced to a very simple form: piecewise-constant.

A fundamental problem in climate research is that of explaining how the Earth’s climate remains stable on very long time scales. Positive feedback mechanisms such as the ice-albedo feedback and the lower tropospheric water vapour/infrared radiative feedback on sea surface temperature perturbations are known to exist which could, in principle, drive the climate system far from its observed mean state even in the absence of any external forcings. Extreme scenarios that have been envisaged are a completely ice-covered earth on the one hand and a runaway greenhouse such as appears to have occurred on Venus on the other.

We examine linear waves on the beta-plane over topography which consists of isolated radially-symmetric irregularities. We assume that the radii of those are much smaller than the characteristic distance between neighbouring topographic features, and that the latter parameter is much smaller than the wavelength.

We study the evolution of initially axisymmetric, large scale (Bu ≪ 1), geostrophic (Ro ≪ 1), barotropic vortices. In such a parameter regime, asymptotic expansion of rotating shallow water (RSW) leads to the frontal geostrophic (FG) model which is numerically implemented. Both in FG and in RSW, anticyclones travel westward faster than cyclones. In FG, vortices of both signs show initially a secondary meridional northward drift which we detail. However this last feature is not present in RSW numerical simulations. Vortices behave like quasi-geostrophic (QG) vortices i.e, showing an initial meridional drift that depends on vortex sign.

Coastal orography can impose distinct imprints upon the structure and dynamic behavior of the marine atmospheric boundary layer (MABL). Abrupt changes in MABL depth, wind speed, and MABL inversion strength have been noted in several over ocean field experiments adjacent to mountainous coastlines. These variations, which occur over much smaller spatial scales than that of the synoptic forcing, tend to be particularly pronounced in the vicinity oftopographic points and capes. Regions of large stress, and curl of stress, frequently appear in the lee of such coastal protrusions. Mesoscale mountain gravity waves, diurnally varying cross-coast baroclinity and supercritical flow effects can all contribute to these pronounced, localized gradients.

Thermocline processes likely play an important role in climate variability at decadal timescales. Surface anomalies subducting at midlatitudes propagate in the main thermocline toward the tropics with a timescale close to decadal. Thermocline variability is also associated with baroclinic Rossby waves, a fundamental agent in the adjustment of the ocean circulation, whose propagation time can introduce a delay of several years in the oceanic feedback to atmospheric forcing.

Abyssal flows, as part of the global thermohaline circulation, make a significant contribution to the flux of heat over the earth, and therefore affect the planet’s climate. In the Atlantic, the deepest flow consists of Antarctic Bottom Water, which originates in the Weddell Sea near Antarctica and flows northward along the western boundary of the Atlantic ocean. While part of this flow recirculates within the Brazil Basin, remaining in the southern hemisphere, part of the flow is observed to cross the equator into the northern hemisphere (DeMadron & Weatherly, 1994; Friedrichs & Hall, 1993).

Several major problems, namely, uncertain surface forcing function, unknown open boundary conditions (OBC), and pressure gradient error using the σ-acoordinate, affect the accuracy of coastal ocean prediction. At open lateral boundaries where the numerical grid ends, the fluid motion should be unrestricted. Ideal open boundaries are transparent to motions. The most popular and successful scheme is the adjoint method. The disadvantages that may restrict its use are ocean-model dependency and difficulty in deriving the adjoint equation when the model contains rapid (discontinuous) processes, such as change of ocean mixed layer from entrainment to shallowing regime. Development of a ocean-model independent algorithm for determining the OBC becomes urgent.

The South China Sea (SCS), Yellow/EaBt China Sea (YES), and Japan/East Sea (JES) are major east Asian marginal seas (EAMS). The complex topography includes the broad shallows of the Sunda Shelf in the south/southwest of SCSi the continental shelf of the Asian landmass in the north, extending from the Gulf of Tonkin to the YES; a deep, elliptical shaped SCS and JES basins, and numerous reef islands and underwater plateaus scattered throughout (Fig. 1a). The shelf that extends from the Gulf of Tonkin to the YES is consistently near 70 m deep, and averages 150 km in width.

FASTEX (Fronts and Atlantic Storm Track Experiment) took place in January and February of 1997. The main purpose of the experiment was to observe in detail the evolution of weather systems crossing the Atlantic using airborne Doppler radar and dropsonde observations as well as an enhanced coverage of conventional and ship observations (Joly et al., 1997).

Low-order models (LOMs) are commonly developed by employing the Galerkin technique. Unfortunately, along with a number of highly attractive features, the method does not provide criteria for selecting modes, nor a guarantee that a model based on a particular set of modes will behave anything like the original system. Moreover, fundamental conservation properties of the fluid dynamical equations are sometimes violated in LOMs, and they may exhibit unphysical behavior (throughout the paper conservation is assumed in the absence of forcing and dissipation).

The generation of instability in inviscid, non-diffusive geophysical flows is generically caused by a resonan ce between two wave modes. The weakly nonlinear unfolding of this situation is described in the long-wave regime, using a particular two-layer quasi- geostrophic model as an illust rative example. The outcome is a system of two coupled Korteweg-de Vries equations. This system contains a very rich solution set, consisting typically of solit ary wave interactions. We will describe some numerical solutions of the coupled Korteweg-de Vries equations, supplemented by perturbation analyses. We also report on some preliminary analogous numerical simulations of the full two-layer quasi-geostrophic system.

The Stommel and Arons (1960 a,b) model of the circulation in the abyssal ocean is modified by allowing the temperature (homogeneous in Stommel — Arons) to vary with depth and latitude and weakly with longitude. The temperature and vertical velocity are specified at the top of the abyss as functions of latitude and weakly of longitude. The resulting distributions of temperature for the North Atlantic show agreement with the climatological data given in Levitus (1982) when the vertical coefficient of thermal diffusion is 125 × 10−4m2s−l. The vertical, meridional and zonal components of velocity show significant change in structure from the Stommel — Arons expressions.

Available pot ential energy (APE) in a compressible ocean is defined as the sum of available gravitational pot ent ial energy (AGPE) and available int ern al energy (AIE ). It is shown that AGPE in a compressible ocean is 40 t imes larg er than that calculate d from the t radit ional quasi-geostrophic approximation. In addition, there is also a large amount of AIE. The major sources of AGPE are due to sur face cooling and vert ical mixing in the interior. Cooling at high latitude also leads to a decrease in AlE and an increase in AGPE, and thus induces a maj or energy transformation in the compressible ocean. In comparison, the role played by haline forcin g is minor.

It is shown that, if wind stress is balanced by form stress in a zonal and depth integral of the zonal momentum equat ion, at each latitude, then wind stress curl is balanced by bottom pressure torque in a zonal integral of the barotropic vorticity equation. It is, therefore, unclear that viscosity is important in western boundary cur rents.

In this contribution we present a formulation of quasi-geostrophic potential vorticity (QGPV) for a generalised vertical coordinate, η. The original motivation for this work was in the possible use of PV as a control variable for variational data assimilation (Var) which would require the inversion of PV on model levels. In addition, we are unaware of any previous formulation of the QG system for a general vertical coordinate and present the following as a contribution to the literature on the subject. We find that our formulation is strictly appropriate only for vertical coordinates that are not terrain-following. Ways of overcoming this limitation are suggested.

This study analyzes the response of t he ocean thermocline to a density front at the sea sur face. A set of comput at ions has been carrie d out, solving the thermocline structure that results from a sur face density front and analyzing the behavior of the solut ion as t he location of the front at the seasur face varies in the meridional direction. Infact, though theoret ical studies often assumes a constant SSD (Sea Sur face Density) gradient, a recent observat ional study (Raffaele and Rudn ick, 2000) suggests that much of the sur face density variat ion, even on large scales, is gat hered into frontal gradients.

This study is motivated by the need of explaining the formation of a class of common but small atmospheric disturbances known as non-supercell tornado (NST). Their properties have been well documented (e.g. Wakimoto and Wilson, 1989). The local environment that spawns a NST typically has a significant horizontal shear, a boundary outflow, a weak vertical shear and is weakly stratified. It is hypothesized that NST could arise from an instability of a horizontal shear flow with respect to non-hydrostatic columnar disturbances in the presence of implicit buoyancy related to a boundary outflow. The objective of this study is twofold. One is to delineate the nature of non-hydrostatic barotropic instability mechanism per se. The other is to ascertain the extent to which the working hypothesis is valid.

Global atmospheric models are usually formulated upon latitude-longitude grids. Near the poles, these grids have disproportionately high resolution, which may severely constrain the time step of integration or require special filtering. Advection problems may also occur near the poles of such grids. Quite recently, a global conformal-cubic grid was devised by (1996). This grid avoids the disadvantages of latitude-longitude grids, but does require careful selection of numerical techniques to account for the eight vertices of the grid (McGregor, 1996).

Whether concerned with the very small, as in elementary particle physics, or with the very large, as in cosmology, relativistic quantum theories are at the forefront of modern physics. Since the unifying principles of physics are symmetry considerations, their validi ty can not depend on specific numerical values, i.e., they can not rely on arbitrary concepts of small and large or fast and slow. If one theory can be proved to be mathematically analogous to another, their solutions must be the same independent of physical scales. In that spirit, sp ecial relativity in the widest sense can be defined as compliance with causality under the assumption that the interaction be tween events in spacetime is mediated at finit e speed. Similarly, quantum mechanics on a basic level, simply means a transition away from deterministic force laws, or dynamics in the classical sense, to wave mechanics in which kinematical properties such as energy and momentum play the most important role.

Density-driven benthic flows are important in the dynamics of marginal seas, river estuaries and other coastal regions (LeBlond et al., 1991; Price & O’Neil Baringer, 1994). They often occur along sloping continental shelves, flowing with shallower water on their right (in the northern hemisphere). Mesoscale gravity currents, which are to be discussed in this study, arise from a geostrophic balance between down-slope acceleration due to gravity and the Coriolis force, while their dynamics is characterized by lengthscales on the order of the Rossby deformation radius. There is mounting evidence that such flows are subject to instability, which may drastically alter the mean flow and culminate in a series of isolated plumes or eddies (Armi & D’Asaro, 1980; Houghton et al., 1982).

The properties of solitary topographic Rossby waves (modons) in a uniformly rotating two-layer ocean over a constant slope are analyzed. The modon is described by exact, form preserving, uniformly translating, horizontally localized, nonlinear solution to the inviscid quasigeostrophic equations. Baroclinic modons over topography are found to translate steadily along contours of constant depth in both directions: either with negative speed (within the range of the phase velocities of linear topographic waves) or with positive speed (outside the range of the phase velocities of linear topographic waves). The lack of resonant wave radiation in the first case is due to the orthogonality of the flow field in the modon exterior to the linear topographic wave field propagating with the modon translation speed that is impossible for barotropic modons. Another important property of a baroclinic topographic modon is that its integral angular momentum must be zero only in the bottom layer; the total angular momentum can be non-zero unlike for the beta-plane modons over flat bottom. This feature allows for modon solutions superimposed by intense monopolar vortices in the surface layer to exist.

Pressure (isobaric or p-) coordinates in non-hydrostatic (NH) dynamics with pressure in the role of the vertic al coordinate were first introduced by (1974), and generalized to sigma-coordinate framework by (1984). First numerical applicat ions of the NH model in pressure related coordinates where developed by (1991) for twodimensional case, and by (1992) for three-dimensional motion.

Low frequency oscillations of the atmosphere with periods of 30-60 days have been discovered in Tropics and later in the Northern Hemisphere (Madden & Julian, 1971; Anderson & Rosen, 1983). This phenomenon has been addressed by many investigators in attempts to clarify the physics behind it. A related and even broader issue is the low-frequency variability (LFV) of the Earth’s atmosphere which is frequently associated with the existence of multiple, systematically recurring, persistent states observed in the real atmosphere (Mo & Ghil, 1988; Yang & Reinhold, 1991; etc.).

A better understanding and prediction of sea-surface temperature (SST) is one of today’s key resear ch areas in climate variability. Results present ed here focus on an ocean model with special emphasis on the Indo-Pacific Oceans, including th e Indonesian throughflow. Three main improvements contribute to th e state-of-t he-art perform ance of the model: a hybrid mixed-layer scheme for the upp er boundary layer of the ocean, a parameterization for t idal mixing in the Indonesian archipelago, and an atmospheric boundary layer model to calculate improved heat and freshwat er fluxes for ocean-only models. Th e result ing pat tern s of mixed-layer depths, water mass tr ansformation s due to tidal mixing in the Indonesian Seas and heat flux anomalies are all in good agreement with observations.

From the practical point of view the long-range weather forecasting (LRWF) is one of the most important problems in meteorology. Although the predictability theory gives the indication that day-to-day weather variations can be forecasted well for about threefour weeks ahead, in reality the predictability limit of such weather variations is about one week at present. Moreover, because of interactions of short- and long-term weather variations the predictability limit for temporally averaged weather characteristics is almost of the same value. Instabilities of atmospheric processes is the main reason for these limitations, and meteorologists are therefore trying to reduce these instabilities in models in order to represent the long-term weather variations as direct responses of the atmosphere to boundary forcing and thereby enhance the predictability of these longterm variations. Unfortunately, it is not possible to reach this goal without the loss of the qualitative similarity between the modeled and real-world atmospheric dynamics.

Spatio-temporal evolution of meanders on a baroclinically unstable jet over a topographic slope is investigated using pulse asymptotics and numerical simulations. An unperturbed jet is prescribed by a potential vorticity front in the upper layer overlaying intermediate layers with weak potential vorticity gradients and a quiescent bottom layer over a positive (same sense as isopycnal tilt) cross-stream topographic slope.An initially localized meander evolves into a wave packet growing and propagating downstream. The pulse asymptotics oflinear waves allows to characterize the structure of amplifying baroclinic wave packets by spatio-ternporal modes which grow exponentially along some rays x/t=const but decay along other rays. In a fully nonlinear numerical solution the instability growth is compensated by the nonlinear terms and the central part of wave packet saturates. The upstream and downstream development ofthe disturbance near the leading and trailing edges ofthe wave packet obeys the linear wave theory.For a weak bottom slope of0.002 the growth rate is only 10% less than that for a flat bottom. Nevertheless, meanders over a flat bottom are able to pinch offresembling warmand cold-core rings, while in the presence ofa weak bottom slope the maximum amplitudes of meanders and associated deep eddies saturate without eddy shedding.Two physical mechanisms are important to understand the effects of topographic slope: It efficiently controls the nonlinear meander growth via constraining the development ofassociated deep eddies. The bottom slope modifies the evolution ofdeep eddies and causes their phase displacement in the direction ofthe upper layer troughs /crests, thus limiting growth ofthe meanders.Behind the wave packet deep eddies form a nearly zonal circulation which stabilizes the jet. The main equilibration mechanism is homogenization ofthe lower layer potential vortic ity by deep eddies.

The purpose of this contribution is to very briefly describe some simulations we have done concerning the evolution of modal disturbances to a planar jet in which the phase velocity of the perturbation is initially equal to the maximum jet velocity. For a more complete discussion of this work see Swaters (1999,2000). As is well known, the perturbation stream function for this configuration is algebraically singular at the jet maximum unlike the logarithmic singularity of a critical layer in a monotonic shear flow.

WKB theory has been the main analytical tool to investigate the effects of a slowlyvarying topography on barotropic and baroclinic Rossby waves in a horizontally uniform stratified ocean for over three decades now, starting with the pioneering work of (1969). During this period, the main focus was on understanding how the topography modifies the vertical structure of the linear normal modes and the local dispersion relationship (e.g., (1994) and references therein), with little or no attention for their propagation speed and direction. This latter issue was tackled only recently by (1999) (KB99 thereafter), who computed rays and wave amplitudes for baroclinic Rossby waves propagating over realistic topographies (albeit considerably smoothed ones) for the five ocean basins, motivated by the recent analysis of TOPEXlPoseidon altimeter data by (1996). Although KB99’s main result is that large scale topography little impacts the phase speeds and wave amplitudes of first-mode baroclinic Rossby waves (compared with standard, linear, normal -mode theory), in (2000) (TMC hereafter), we questioned the assumption of no scattering between WKB modes upon which KB99 ‘s calculations rely. The support for our claim is illustrated in Fig. 1 which compares the amplitude of annual-period baroclinic Rossby waves excited along an eastern boundary computed by a direct numerical method (left panel) and by means of WKB theory as in KB99 (right panel).

There are many applications in atmosphere and ocean science that require the calculation of sensitivities. Examples include variational data assimilation [7], parameter estimation [9], initialization of ensemble forecasts [1], and finding sensitive areas for the purpose of targeting observations [6]. Sensitivity calculations can often be accomplished efficiently using the adjoint of a tangent linear model, since the computational cost of a single integration of an adjoint is typically no more than a few times the cost of a single integration of its parent forward model.

A prominent feature of two-dimensional and quasi-geostrophic turbulence is the formation of large-scale coherent structures among the smallscale fluctuations of the vorticity field. This separation-of-scales behavior is a consequence of the conservation of both energy and enstrophy by the dynamics, which results in a net flux of energy toward large scales and a net flux of enstrophy toward small scales. Many flows of this kind, whether free-decaying flows or weakly driven, can therefore be described approximately as coherent, deterministic structures on the large scales and disorganized, random motions on the small scales.

Bottom topography influences the oceanic dynamics in a variety of ways, depending on its height, spatial scale, and on the flow regime. A physically relevant situation, which can be partly investi gated analytically, is that of a topography whose scale is much smaller th an th e typi cal scale of th e moti on. Assuming periodic or random topography and small-amplitude flow, th e standard technique of homogenizat ion can be appli ed to derive averaged, or homogenized, evolution equations for the large-scale flow. In such equations, th e usual (small-scale) topographic ter ms are replaced by non-tr ivial averaged terms which account for the large-scale effect of topography. The homogenization approach can thus be interpreted as providing an asymptotically consistent parameterization of topography in large-scale models.

Time variations of the atmospheric state have two periodic components known as a diurnal cycle and an annual cycle, which are periodic responses to the periodic variations of the external forcings due to the earth’s rotation and revolution, respectively. Intraseasonal and interannual variations are defined as deviations from the periodic annual response; generally the intraseasonal variation means lowfrequency variation with week-to-week or month-to-month time scales, while the interannual one means year-to-year variation. Some part of these variations is a response to the time variations of the external forcings or boundary conditions of the atmospheric system, while the rest is generated within the system. Well known forcings with interannual time scales are ll-year solar cycle, irregular and intermittenteruptions of volcanos and interannual variations of sea surface temperature, although the last one should be considered as internal variation of the coupled system of the atmosphere and oceans. Quasi-Biennial Oscillation(QBO) of the mean zonal wind in the equatorial stratosphere, which is basically caused by the interaction between the mean zonal wind and equatorial waves propagated from the troposphere, could be considered as variation of a lateral boundary of the mid-latitude stratosphere. On the other hand, external forcings are not so significant in the intraseasonal time scales; fundamentally intraseasonal variations could be considered as internal one which may exist even under constant external conditions.