Eric Delhez

The concept of age in marine modelling I. Theory and preliminary model results

The age of a particle of a seawater constituent is defined to be the time elapsed since the particle under consideration left the region, in which its age is prescribed to be zero. An Eulerian theory of the age is presented, in which advection, diffusion, production and destruction phenomena are properly accounted for. The key hypothesis is that the mean age of a set of particles is to be evaluated as the mass-weighted average of the ages of the particles under study. The basic variable is the concentration distribution function, representing, at a given time and location, the distribution over the age of the concentration of the constituent being considered. This function satisfies a partial differential equation, which, upon appropriate integration over the age, yields the equations, in flux form, governing the evolution of the concentration and the age concentration. The ratio of the latter variable to the former is the mean age. Further theoretical developments are presented, including a thought experiment showing that mixing processes cause the ages of various constituents to be different from each other. The potential of the age as a tool for understanding complex marine flows is briefly demonstrated by analysing the results of two numerical models. The ages of a passive tracer, a radioactive tracer and the water are computed, along with a suitably defined radio-age. First, the fate of tracers released into the English Channel at La Hague is simulated. Then, ages are computed in the World Ocean as a measure of the time that has elapsed since leaving the surface layers. A theorem is demonstrated, which specifies that the age of the radioactive tracer must be smaller than the relevant radio-age, the latter being smaller than the age of the passive tracer, which, under appropriate hypotheses, can be seen to be equivalent to the age of the water. These inequalities seem to be remarkably robust, since they are found to hold valid in most of the numerical and analytical results examined in the present study. On the other hand, a dimensionless number is highlighted, which is believed to play an important role in the scaling of the differences between ages

The concept of age in marine modelling II. Concentration distribution function in the English Channel and the North Sea

The age of seawater and the age of real or idealized tracers are often used as diagnostic tools to better understand complex hydrodynamic flows. In most studies, the focus is on some averages of the ages of the different particles making up a water parcel. The theory developed in Delhez et al. [Ocean Modell. I (1999) 17] and Deleersnijder et al. [J. Mar. Syst. 28 (2001) 229] provides, however, a more detailed description of the distribution of the ages of these particles through the so-called concentration distribution function. In this paper, the numerical aspects of the resolution of the evolution equation for the concentration distribution function in a five-dimensional space (time x 3D space x age dimension) are developed. Evolution equations for the moments of the concentration distribution function up to any order are also derived. A real case application of this theory to the simulation of the advection-dispersion of tracers (technetium-99) discharged at the nuclear fuel reprocessing plant of Cap de La Hague is described. The comparison of the results with those from previous studies demonstrates the advantages of the new method for the computation of the mean age. In particular, the method provides a detailed description of the temporal variations of the mean age only. The analysis of the full concentration distribution function and of its basic statistics shows that the standard deviation of the age of the different particles is far from negligible and should never be overlooked when analyzing age fields. A simplified analytical example suggests that the standard deviation of the age distribution is a measure of the integrated diffusion undergone by the tracer along its path from the source to the observation point

A note on the age of radioactive tracers

wThe age of a water mass is often estimated experimentally using the radio-age computed from the distribution of a radioactive tracer (radiocarbon, helium-tritium). Deleersnijder et al. [J. Mar. Syst. 28 (2001) 229.] have shown that the radio-age underestimates the age of the water and is larger than the age of the radioactive tracer used for its evaluation. This result is generalized here to radio-ages computed from the ratio of two radioactive tracers. The differences between the different ages are also studied analytically and numerically as functions of the decay rate of the radioactive tracers. For small decay rates, the difference between the age of the water mass and the radio-age is shown to be proportional to the decay rate. It depends also on the level of mixing in the system; even radioactive tracers with small decay rates can provide poor estimates of the age of the water mass in a strongly diffusive flow. For small half lives, both the radio-age and the age of radioactive tracers decrease as the inverse of the square root of the decay rate. The same analysis applies to some extent to the estimates of the age of a water mass from stable tracers with known time dependent sources (e.g. chloroflurocarbons)

Some properties of generalized age-distribution equations in fluid dynamics

The concept of age in fluid dynamics is analyzed in the case of a tracer advection-diffusion equation. From the general solution in a uniform velocity field, it is shown that unexpected symmetry properties arise for the age field. In particular, for a point release, the age field is isotropic, regardless of the direction of the ow and the value of the diffusion coefficient. The analysis is then extended to situations with time-varying currents, where the symmetry can be broken under some circumstances. Finally, we show a method by which a time-dependent problem can be used to assess a stationary concentration distribution function, providing details about the propagation of younger and older material at a given location.

On the residual advection of passive constituents

Abstract In this paper, the different approaches used to simulate the long-term advection of passive constituents on tidal shelves are discussed in the framework of large scale hydrodynamic modelling. The direct approach in which the unsteady mesoscale (tides and storm surges) and macroscale (monthly or seasonal mean processes) currents are used is very demanding in computer resources (CPU and memory). On the other hand, the use of residual velocities gives a larger understanding of the long-term transport processes and simplifies numerical treatments. However, the definition of such appropriate velocities is still an open question. In the context of large scale models, Lagrangian residuals are not applicable. Eulerian residual transport velocities fail to represent long-term motions when tidal non-linearities are important. The first order Lagrangian residual velocity introduced by Feng et al. (1986) and generalized here is shown to be a very good solution. The North-Western European Continental Shelf (NWECS) is used as an example to compare the numerical solutions obtained with the different approaches.

Preliminary results of 3-D baroclinic numerical models of the mesoscale and macroscale circulations on the North-Western Europena Continental Shelf

Abstract Two mathematical models for the study of mesoscale (tides, storm surges…) and macroscale (general circulation…) motions are described. Both models are derived from the GHER 3-D baroclinic turbulent closure primitive equation model and adapted to the specific spectral windows of, respectively, mesoscale and macroscale processes. Numerical methods are discussed: 3-D finite differences, mode-splitting, σ-transform… The mesoscale model is applied to the simulation of the M2 tide on the North-Western European Continental Shelf. The residual circulation model is based on time averaged primitive equations which explicitly include mesoscale Reynolds stresses that are computed by the mesoscale model. Preliminary results of the application of this model in typical summer conditions are presented.

Transient behaviour of water ages in the World Ocean

The transient behaviour of the age of the water and that of the surface water is studied in the World Ocean. At any time and position, the age of the water, the age and the concentration of the surface water are seen to obey a simple algebraic relation. The latter is illustrated by means of results of a three-dimensional World Ocean model and the analytical solution of an idealised, purely-diffusive, one-dimensional problem.

Macroscale ecohydrodynamic modeling on the Northwest European Continental Shelf

Abstract A 3D coupled hydrodynamic–biological model is applied to the simulation of the biological processes on the Northwest European Continental Shelf. The model operates in the macroscale spectral window (time scales of a month or a season) without explicit description of the higher frequency processes but with an adequate modeling of their influence on larger time scales. The hydrodynamic sub-model is 3D, baroclinic and includes a refined turbulence closure. The non-linear interactions of mesoscale fluctuations are described by means of the generalized mesoscale Reynolds stresses and the Stokes drift transport velocity field. The biological sub-model describes the nitrogen and carbon cycles through the food web with 17 state variables representing 9 compartments: inorganic nutrients, small phytoplankton, large phytoplankton, dissolved organic matter, pelagic bacteria, heterotrophic flagellates, pelagic detritus, zooplankton and benthic organic detritus. The simulation emphasizes the strong influence of the local depth and of the stability of the water column on the whole annual cycle of phytoplankton. In well-mixed shallow areas, the chlorophyll concentration increases in early spring and the primary production occurs steadily until October. In deeper areas, the spring bloom is much sharper and appears later, after the set up of the seasonal stratification. The main part of the primary production happens during the bloom period. The macroscale approach provides results that are comparable with observations and results of other more classical models describing explicitly the higher frequency processes. The current method allows, however, a more direct insight into the dynamics of the system and into the interactions between the hydrodynamics and the biology. Also, it greatly reduces the CPU requirements and is therefore particularly suited for repeated or long term simulations.

A spline interpolation technique that preserves mass budgets

Abstract The usual practice of forcing budget models by linear interpolations of mean data does not produce a forcing whose mean is the data value required. The usual third-order spline is modified into a fourth-order spline, called mc-spline, to cope with this issue. The technique provides a smooth and faithful continuous interpolation of the original data that is well suited for its graphical representations or for the forcing of numerical models.

Dissection of the GHER turbulence closure scheme

Abstract In this paper, the turbulence closure scheme implemented in the GHER hydrodynamic model is described in detail. Two case studies carried out in two contrasting conditions—one in the shallow, tide dominated, north-western European continental shelf, and the other in the deep Mediterranean Sea—are used to identify the dominant terms of the equation for the turbulent kinetic energy, first theoretically, secondly from the results of the simulations. In both domains, the dominant terms are the local destruction and production terms, the vertical diffusion term and to a smaller degree, the time derivative. Advection and horizontal diffusion turn out to be negligible in most of the relevant cases for such large scale studies. This opens the way to simplifications and optimisations of the numerical models.