**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 m^{2}/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 ^{7}Be, 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 m^{2}/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 ^{7}Be 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 R_{g} 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 R_{g} 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.

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