(last updated May 2005)
We examine the accuracy, computational advantage and feasibility of using an ``offline" approach for modeling ocean tracers in eddy-resolving calculations. In the offline approach time-averaged flow-fields and mixing coefficients, stored during a prior online run, are re-used leaving only the tracer equation to be integrated. We find that, provided the flow fields used to drive the offline calculation are averages on time-scales close to or below the inertial period, the tracer distributions depart little from the reference online case. In addition we find that the offline timestep, no longer limited by dynamical constraints, can be increased by almost an order of magnitude relative to the online value. These two factors enable close offline reproduction of an online solution in a fraction of the time promoting the use of offline modeling in eddy resolving tracer models.
H.L. Jones, C.N. Hill, M.J.Follows, S.W. Dutkiewicz.
manuscript submitted to Ocean Modeling May 2005
Effective diffusivities assocoated with geostrophic eddies in the southern ocean are estimated by numerically monitoring the lengthening of idealized tracer contours as they are strained by surface geostrophic flow observed by satellite altimetry. The resulting surface diffusivities show considerable spatial variability and are large (2000 m2s-1) on the equatorward flank of the Antarctic Circumpolar Current and are small (500 m2s-1) at the jet axis. Regions of high and low effective diffusivity are shown to be collocated with regions of, respectively, weak and strong isentropic potential vorticity gradients. The maps of diffusivity are used, along with climatological estimates of surface wind stress and air-sea buoyancy flux, to estimate surface meridional residual flows and the relative importance of Eulerian and eddy-induced circulation in the streamwised-average dynamics of the Antarctic Circumpolar Current.
J.C. Marshall, E.F.Schuckburgh, H.L. Jones, C.N. Hill.
manuscript submitted to J. Phys. Oceanogr March 2005
This work, which grew out of process studies of oceanic deep convection (see below), turned the ocean deep convection problem on its head to investigate the equilibration of a warm lens through its baroclinic instability. It is published as
Deep convection is an integral part of the large-scale meridional ocean circulation. As such it is a process which must exert a fundamental influence on the global climate. The ocean's response to large-scale surface buoyancy loss is to overturn in narrow "chimneys" only a few tens of kilometres wide and therefore too small to be resolved in global-scale climate models. Consequently, in the large-scale models used to study climate, deep convection has to be parameterised. To ensure prognostic confidence in a climate model (particularly in view of what has been found to be the extreme sensitivity of the global thermohaline circulation to the parametric scheme employed) it is essential that the parameterisation scheme should have a firm physical foundation.
In my work I use a general circulation model (the MITgcm) to perform idealised process studies designed to capture and elucidate the key features of deep convection. My earliest work concerned plume and eddy scale dynamics. More recently I have explored how convectively modified water communicates with the ocean at large.
An important yet poorly understood aspect of the water mass transformation process in the ocean is the manner in which the convected fluid, once formed, is accommodated and drawn in to the general circulation. Following "violent mixing" in the open ocean that creates a deep homogeneous body of fluid, restratification of the surface (around 500m) layer is observed to occur rapidly, sealing over the convection patch. Recent hydrographic casts and tomography inversions in the Gulf of Lions by Send et al., for example, show that very quickly, within a week or so of the cessation of cooling, a stratified near-surface layer develops on top of the mixed patch. This restratification occurs much more rapidly than can be accounted for by air-sea fluxes. By analytical and numerical study the authors argue, that advection by geostrophic eddies spawned by the baroclinic instability of the mixed patch is likely to be a principal mechanism by which restratification occurs. A restratification timescale,
where r is the radius of the patch of mixed water, h its depth, and N, the ambient stratification, can be deduced from the magnitude of the lateral buoyancy flux asscoiated with the geostrophic eddy field. This formula finds support from numerical results and is in broad agreement with the observations. Finally the results of the study are used to interpret recent field observations in the Labrador and Mediterranean Seas.
A solution strategy for the incompressible Navier Stokes equations (INS) is outlined which renders ocean models based on them competitive with those that assume hydrostatic balance on all scales. It is shown that non-hydrostatic models rooted in INS can be designed which, when deployed in the hydrostatic limit, require no more computational effort than models based on the hydrostatic primitive equations (HPE). But unlike HPE, the model can also be used to address convective, non-hydrostatic scales if its resolution is increased. The ideas are illustrated in simulations of convection, baroclinic instability and large-scale circulation in the Mediterranean.
A simple point-vortex "heton" model is used to study localised ocean convection. In particular, the statistically steady state that is established when lateral buoyancy transfer, effected by baroclinic instability, offsets the localized surface buoyancy loss. These predications compare favorably with the values obtained through numerical integration of the heton model.
The steady state of the heton model can be related to that in other convection scenarios considered in several recent studies by means of a generalized description of the localised convection. This leads to predictions of the localisedequilibrium density anomalies in these scenarios, which concur with those obtained by other authors. Advantages of the heton model include its inviscid nature, emphasiising the independence of the fluxes affected by the baroclinic eddies from molecular processes, and its extreme economy, allowing a very large parameter space to e covered. This economy allows us to examine more complicated forcing scenarios: for example, forcing regions of varying shape. By increasing the ellipticity of the forcing region, the instability is modified by the shape and, as a result, no increase in lateral fluxes occurs despite the increased perimeter length.
The parameterisation of convective mixing by a redistribution of potential vorticity, implicit in the heton model, is corroborated; the heton model equilibrium state has analogous quantitative scaling behaviour to that in models or laboratory experiments that resolve the vertical motions. The simplified dynamics of the heton model therefore allows the adiabatic advection resulting from baroclinic instability in controlling the properties of a water mass generated by localised ocean convection. A complete parameterisation of this process must therefore account for fluxes induced by horizontal variations in surface buoyancy loss and affected by baroclinic instability.
An initially resting ocean of stratification N is considered, subject to buoyancy loss at its surface of magnitude Bo over a circular region of radius r, at a latitude where the Coriolis parameter is f. Initially the buoyancy loss gives rise to upright convection as an ensemble of plumes penetrates the stratified ocean creating a vertically mixed layer. However, as deepening proceeds, horizontal density gradients at the edge of the forcing region support a geostrophic rim current, which develops growing meanders through baroclinic instability. Eventually finite-amplitude baroclinic eddies sweep stratified water into the convective region at the surface and transport convected water outward and away below, setting up a steady state in which lateral buoyancy flux offsets buoyancy loss at the surface. In this final state quasi-horizontal baroclinic eddy transfer dominated upright "plume" convection.
By using "parcel theory" to consider the energy transformations taking place, it is shown that the depth, h(final), at which deepening by convective plumes is arrested by lateral buoyancy flux due to baroclinic eddies, and the time t(final) it takes to reach this depth, is given by
both independent of rotation. Here gamma and beta are dimensionless constants that depend on the efficiency of baroclinic eddy transfer. A number of laboratory and numerical experiments are then inspected and carried out to seek confirmation of these parameter dependencies and obtain quantitative estimates of these constants. It is found that gamma equals 3.9 plus or minus 0.9 and beta equals 12 plus or minus 3.
Finally, the implications of our study to the understanding of integral properties of deep and intermediate convection in the ocean are discussed.
The intensity and scale of the geostrophically adjusted end state of the convective overturning of a homogeneous, rotating ocean of depth H at a latitude where the Coliolis parameter is f, induced by surface buoyancy loss of magnitude Bo, are studied by numerical experiment. The experiments are related to observations and laboratory studies of open-ocean deep convection. A numerical model based on the nonhydrostatic Boussinesq equations is used. The grid spacing of the model is small enough that gross aspects of the convective plumes themselves can be resolved yet the domain of integration is sufficiently large to permit study of the influence of plumes on the large scale and geostrophic adjustment of the convected water.
Numerical simulations suggest that cooling at the sea surface is offset by buoyancy drawn from depth through the agency of convective plumes. These plumes efficiently mix the water column to generate a dense chimney of fluid, which subsequently breaks up through the mechanism of baroclinic instability to form spinning cones of convectively modified water that have a well-defined and predictable scale.
A measure of the importance of rotation on the convective process is provided by a natural Rossby number introduced by Maxworthy and Narimousa:
where l_rot is the length scale that marks the transition from three-dimensional, thermally driven turbulence to quasi-two-dimensional, rotationally dominated motions. Here u_rot is the velocity of a particle gyrating in inertial circles of radius l_rot.
In the parameter regime typical of open-ocean deep convection, we find that Ro is less than or equal to 1; rotation influences the intensity and scale of both plumes and cones. In particular, the scale, intensity, buoyancy excess, and generation rate of the cones of geostrophically adjusted fluid, which result from the breakup of the chimney, are found to depend in a predictable way on the single nondimensional number, formed from the external parameters f, Bo and H.
It is argued that non-hydrostatic effects are important in the dynamics of opne-ocean deep convection on horizontal scales of around 1km, typical of sinking plumes. The formulation and numerical implementation of a non-hydrostatic ocean model appropriate for the explicit representation of plume-scale dynamics is described. The model is used to study the spin-up, through convective overturning induced by surface cooling, and subsequent geostrophic adjustment of a baroclinic vortex with a horizontal scale of the order of the Rossby radius of deformation.
BACK to the top Jones, H.L., C.N. Hill, M.J. Follows and S.W. Dutkiewicz, 2005. Is there a computational advantage to offline tracer modeling at very high resolution? Ocean modeling (submitted). Marshall, J.C., E.F. Schuckburgh, H.L. Jones and C.N. Hill, 2005. Estimates and Implications of Surface Eddy Diffusivity in the Southern Ocean Derived From Tracer Transport. J. Phys. Oceanogr (submitted). Marshall, J.C., H.L. Jones, R. Karsten and R. Wardle, 2002. Can Eddies Set Ocean Stratification? J. Phys. Oceanogr. Vol 32, No 1, January 2002 (26-38). Karsten, R., H.L. Jones and J.C. Marshall, 2002. The Role of Eddy Transfer in Setting the Stratification and Transport of a Circumpolar Current. J. Phys. Oceanogr, Vol 32, No 1, January 2002 (39-54). Jones, H.L. and J.C. Marshall, 1997. Restratification after deep convection. J. Phys. Oceanogr, Vol 27, No 10, October 1997 (2276-2287). Marshall, J.C., H.L. Jones and C.N. Hill, 1996. Efficient ocean modeling using non-hydrostatic algorithms. J. Mar. Sci. Legg, S.A., H.L.Jones and M. Visbeck (1996). A heton perspective of baroclinic eddy transfer in localized open ocean convection. J. Phys. Oceanogr, Vol 26, No 10, October 1996 (2251-2266)