Advection-Diffusion Equations and Turbulence
File last modified 2 November 1998
15.4 The Numbers Game
Before going on to talk about specific oceanographic results, we will digress to discuss various NUMBERS that are of interest in oceanography. Numbers are, by definition, dimensionless quantities which somehow embody physically important characteristics of the systems being studied. More often than not, they indicate the relative importance of various processes, and appear as the ratio of either terms or time scales. The reason for describing your system in terms of these numbers is that often the character of model output tends to depend more on the ratio of terms rather than their absolute value. So, for example, when faced with modeling the behavior of a system over a wide range of velocities and diffusivities, we may find that we need only perform experiments over a range of the ratio of two terms (the Peclet number) rather than doing every conceivable velocity and diffusivity combination. We will list some of the more important, along with typical values and what they mean.
15.4.1 The Reynolds Number
The Reynolds number, often referred to as Re is a measure of the relative importance of intertial to viscous terms. The higher the Reynolds number, the more likely to be turbulent the flow, and the lower the number, the more likely the flow will be laminar. The Reynolds number is defined as
where the denominator is the kinematic viscosity (the greek letter "nu"). This may also be regarded as the ratio of the (molecular) diffusive time scale to the advective time scale. For most fluids (the atmosphere, oceans, etc.) the Reynolds number is of the order of several thousand or greater. Thus most fluids are in a state of turbulent flow.
15.4.2 The Peclet Number
The Peclet number (Pe) is a measure of the relative importance of advection to diffusion. Diffusion here is turbulent diffusion. The higher the Peclet number, the more important is advection. It is given by
and can be arrived at by non-dimensionalizing the advection-diffusion equation. This number may also be thought of as the ratio between the diffusive to the advective time scales. A typical open ocean is characterized by velocities of order .01 m/s, lengths of order 2-3000 km (the size of ocean gyres), and turbulent diffusivities of order 1000 m2/s. This gives a Peclet number of order 20-30.
The trick, though, is in the seemingly arbitrary choice of the length scale L. Clearly, the bigger L becomes, the higher the Peclet number becomes. This is equivalent to saying that given enough time, advection always wins out over diffusion. This is because while the displacement of a particle increases linearly with time with advection, it only increases as the square root of time with diffusion. This can be seen by thinking about diffusion as a random walk experiment (which is what it mathematically is). But it does boil down to this implicit ambiguity that the Peclet number (and hence the apparent relative role of advection and diffusion) depends on the spatial scale of the system being studied.
Radioactive tracers, with their built in decay constants can define their own space scales. This can also be seen by non-dimensionalizing the advective-diffusive-decay equations. The characteristic length scale is the velocity divided by the decay constant, or quite simply the distance a fluid parcel would go before the tracer would be reduced to 1/e of its value by decay. Thus the radiotracer Peclet number would be defined as
Now for a given fluid flow, the length scale will be different for differing radiotracers, so that diffusion and mixing will be more important for one tracer than for another. Consider, for example 7Be, which has a half life of 53.4 days, and thus has a decay probability of 1.51x10-7 s-1. For the subtropical North Atlantic, with velocities of order .01 m/s, and horizontal turbulent diffusivities of order 1000 m2/s, this gives a Peclet number of order 0.7, which says that diffusion and mixing are as/more important than advection. Consider the same situation, however, with tritium (half-life 12.45 years). The same calculation yields a Peclet number of order 50-60, which says that tritium is more affected by advection than diffusion. Now let's turn the problem around and say that if you were interested in studying the effects of diffusion, you'd be more interested in using 7Be than tritium.
15.4.3 The Richardson Numbers
The ocean is in general stably stratified. That is, heavy water is overlain by lighter water. If the reverse were true, then the water column would be gravitationally unstable, and vertical motions (convection) would result that would erase the condition. Now for turbulent displacement to occur vertically in a stratified water column, the fluid particles must overcome the vertical density (buoyancy) gradient. (We'll discuss this more in the next section). Thus one would expect that the ability of the water column to resist this vertical turbulence will be related to the vertical density gradient (also referred to as the rate of buoyancy production). Now one model of the origin of the energy required to produce turbulent motions is the vertical shear in the horizontal velocity. That is, if the horizontal velocity is changing with depth, the different layers traveling at different speeds tend to "rub" against one another, and there must be an overall dissipation occurring to maintain the velocity gradient. This dissipation scales as the square of the velocity gradient, so that defines a Richardson Flux number as
where is the thermal conductivity, is the kinematic viscosity, g is the gravitational constant and u is the horizontal velocity. This is the ratio of buoyancy production to turbulent kinetic energy. Another important quantity is the gradient Richardson number, which is defined by
In situations where Rg decreases much below 1, turbulent diffusion becomes important, and can grow to a point where the system mixes vigorously. Laboratory experiments indicate that a critical Rg of 0.25 is a good approximation for most systems.
15.4.4 Various other Numbers
Various other numbers crop up in different circumstances. Ones that you may hear of are the Prandtl number and the Schmidt number. The former is the ratio of viscosity to thermal diffusion
and the latter is the ratio of viscosity to molecular diffusion
These are often used to compare model or flux calculations between
different situations or chemical species.
GoTo Next Section