Eddy diffusion

Conceptual diagram for a simple derivation of eddy diffusion. An eddy mixes the contents of two fluid regions by injecting streams and filaments back and forth in a quasi-random way. The real process is much more chaotic than a simple spiral suggests. ${\textstyle \Phi _{1}}$ and ${\textstyle \Phi _{2}}$ stand for the concentrations of the same arbitrary substance that is being mixed by the eddy. The length-scale of the two regions influenced by the eddy in this picture is set by the length-scale of the eddy, and not vice versa.

The subsection aims for a simple, rough and heuristic argument explaining how the mathematics of gradient diffusion arises. A more rigorous and general treatment of gradient model is offered in the next subsection, which builds directly on the section on general mathematical treatment (which was not yet assuming gradient model at that early stage and left the covariance of fluctuations as it was). Means are for now not indicated explicitly for maximal simplicity of notation. Also for now neglect the molecular diffusivity ${\textstyle K_{0}}$, since it is usually significantly smaller than eddy diffusivity, and would steer attention away from the eddy mechanism.

Consider two neighbouring fluid parcels with their centers ${\textstyle \Delta x}$ apart. They contain volume concentrations ${\textstyle \phi _{1}}$ and ${\textstyle \phi _{2}}$ of an inert, passive tracer. Without loss of generality, let ${\textstyle \phi _{2}>\phi _{1}}$. Imagine that a single eddy of length scale ${\textstyle \Delta x}$ and velocity scale ${\textstyle U}$ is responsible for a continuous stirring of material among the two parcels. The tracer flux exchanged through the lateral boundary of the two parcels is labelled ${\textstyle J}$. The boundary is perpendicular to the ${\textstyle x}$-axis. The flux from parcel 1 to parcel 2 is then, at least by order of magnitude:

$${\displaystyle {\begin{aligned}J&=\phi _{1}U-\phi _{2}U\\&=-U\Delta \phi \\&=-(U\Delta x){\frac {\Delta \phi }{\Delta x}}\end{aligned}}}$$

This argument can be seen as a physically motivated dimensional analysis, since it uses solely the length and velocity scales of an eddy to estimate the tracer flux that it generates. If the entire studied domain (thought to contain a large number of such pairs ${\textstyle \phi _{1}}$ and ${\textstyle \phi _{2}}$) is much larger than the eddy length scale ${\textstyle \Delta x}$, one can approximate ${\textstyle \Delta \phi }$ over ${\textstyle \Delta x}$ as the derivative of concentration in a continuously varying medium:

$${\displaystyle J=-(U\Delta x){\frac {\partial \phi }{\partial x}}}$$

Based on similarity with Fick's law of diffusion one can interpret the term in parentheses as a diffusion coefficient ${\textstyle K}$ associated with this turbulent eddy, given by a product of its length and velocity scales.

$${\displaystyle J=-K{\frac {\partial \phi }{\partial x}}}$$

using a one-dimensional form of continuity equation ${\textstyle {\frac {\partial \phi }{\partial t}}+{\frac {\partial J}{\partial x}}=0}$, we can write:

$${\displaystyle {\frac {\partial \phi }{\partial t}}={\frac {\partial }{\partial x}}\left(K{\frac {\partial \phi }{\partial x}}\right)}$$

If ${\textstyle K}$ is assumed to be spatially homogeneous, it can be pulled out of the derivative and one gets a diffusion equation of the form:

$${\displaystyle {\frac {\partial \phi }{\partial t}}=K{\frac {\partial ^{2}\phi }{\partial x^{2}}}}$$

This is a prototypical example of parabolic partial differential equation. It is also known as heat equation. Its fundamental solution for a point source at ${\textstyle x=0}$ is:

$${\displaystyle \phi (x,t)={\frac {1}{\sqrt {4\pi Kt}}}\exp {\left(-{\frac {x^{2}}{4Kt}}\right)}}$$

By comparison with Gaussian distribution, one can identify the variance as ${\textstyle \sigma ^{2}(t)=2Kt}$ and standard deviation as ${\textstyle \sigma (t)={\sqrt {2Kt}}\sim t^{1/2}}$, a very typical time dependence for molecular diffusion or random walk.

To conclude this subsection, it described how an eddy can stir two surrounding regions of a fluid and how this behaviour gives rise to mathematics described as "gradient model", meaning that diffusive fluxes are aligned with a negative spatial gradient in concentration. It considered a very simple geometry, in which all variations happen along one axis. The argument used only order-of-magnitude scales of spatial separation and eddy velocity, therefore it was very rough. The next section offers a more rigorous treatment.

This subsection builds on the section on general mathematical treatment, and observes what happens when a gradient assumption is inserted.

Recall the Reynolds-averaged concentration equation:

$${\displaystyle {\frac {\partial \langle \phi \rangle }{\partial t}}+\nabla \cdot \left(\langle {\vec {u}}\rangle \langle \phi \rangle \right)=\nabla \cdot \left(K_{0}\nabla \langle \phi \rangle -\langle {\vec {u}}'\phi '\rangle \right)}$$

We make a similar gradient assumption to that which was motivated in the subsection above with tracer length and velocity scales. However the coefficient value needs not be the same as in the above subsection (which was only specified by order of magnitude). The gradient hypothesis reads:

$${\displaystyle \langle \phi '{\vec {u}}'\rangle =-K({\vec {x}},t)\nabla \langle \phi \rangle }$$

This allows the concentration equation to be rewritten as

$${\displaystyle {\frac {\partial \langle \phi \rangle }{\partial t}}+\nabla \cdot \left(\langle {\vec {u}}\rangle \langle \phi \rangle \right)=\nabla \cdot \left((K_{0}+K)\nabla \langle \phi \rangle \right)}$$

This is again similar to the initial concentration equation, with transformations ${\textstyle \phi \rightarrow \langle \phi \rangle ,{\vec {u}}\rightarrow \langle {\vec {u}}\rangle }$ and ${\textstyle K_{0}\rightarrow K_{0}+K}$. It represents a generalization to Fick's second law (see Fick's laws of diffusion), in presence of turbulent diffusion and advection by the mean flow. That is the reason why down-gradient eddy diffusion models are often referred to as "Fickian", emphasizing this mathematical similarity. Note that the eddy diffusivity ${\textstyle K}$ can in general be a function of space and time, since its value is given by the pattern of eddies that can evolve in time and vary from place to place. Different assumptions made about ${\textstyle K({\vec {x}},t)}$ can lead to different models, with various trade-offs between observations and theory. Sometimes, the term Fickian diffusion is reserved solely for the case when ${\textstyle K}$ is a true constant.[16] ${\textstyle K}$ needs to be at least spatially uniform for it to be possible to write:

$${\displaystyle {\frac {\partial \langle \phi \rangle }{\partial t}}+\nabla \cdot \left(\langle {\vec {u}}\rangle \langle \phi \rangle \right)=(K_{0}+K)\nabla ^{2}\langle \phi \rangle }$$

In this case, the sum of molecular and eddy diffusivity can be considered as a new effective viscosity, acting in qualitatively similar way to molecular diffusivity, but significantly increased in magnitude.

In the context of this article, the adjective"Fickian" can also be used as an equivalent to a gradient model,[17] so a more general form like ${\textstyle K({\vec {x}},t)}$ is permissible. The terminology in scientific articles is not always consistent in this respect.

Gradient models were historically the first models of eddy diffusion.[13] They are simple and mathematically convenient, but the underlying assumption on purely down-gradient diffusive flux is not universally valid. Here are a few experimental counter-examples:

1. For a simple case of homogeneous turbulent shear flow [5] the angle between ${\textstyle -\nabla \langle \phi \rangle }$ and ${\textstyle \langle \phi '{\vec {u}}'\rangle }$ was found to be 65 degrees. Fickian diffusion predicts 0 degrees.
2. On the sea, surface drifters initially farther apart have higher probability of increasing their physical distance by large amounts than those initially closer. In contrast Fickian diffusion predicts that the change in mutual distance (i.e. initial distance subtracted from the final distance) of the two drifters is independent of their initial or final distances themselves. This was observed by Stommel in 1949.[17]
3. Near a point source (e.g. a chimmey), time-evolution of the envelope of diffusing cloud of water vapour is typically observed to be linear in time. Fickian diffusion would predict a square root dependence in time,.[4][7]

These observations indicate that there exist mechanisms different from purely down-gradient diffusion, and that the qualitative analogy between molecular and eddy diffusion is not perfect. In the coming section on statistical models, a different way of looking at eddy diffusion is presented.

Consider an unbounded turbulent flow in which a source at the time ${\textstyle t_{0}}$ determines the scalar field to some value:

$${\displaystyle \phi ({\vec {x}},t_{0})=\phi _{0}({\vec {x}})}$$

${\textstyle {\vec {X}}(t,{\vec {Y}})}$ is the position at time ${\textstyle t_{0}}$ of the fluid particle originating from position ${\textstyle {\vec {Y}}}$ at time t. If molecular diffusion is neglected, ${\textstyle \phi }$ is conserved following a fluid particle. Then, the value of ${\textstyle \phi }$ at the initial and final points of the fluid particle trajectory are the same:

$${\displaystyle \phi ({\vec {X}}(t,{\vec {Y}}),t)=\phi ({\vec {Y}},t_{0})=\phi _{0}({\vec {Y}})}$$

Calculating the expectation of the last equation yields

$${\displaystyle \left\langle \phi ({\vec {x}},t)\right\rangle =\left\langle \phi _{0}({\vec {Y}}(t,{\vec {x}})\right\rangle =\int f_{X}({\vec {x}};t|{\vec {Y}})\phi _{0}({\vec {Y}})d{\vec {Y}}}$$

where ${\textstyle f_{X}}$ is the forward probability density function of particle position.

For the case of a unit point source fixed at location ${\textstyle {\vec {Y_{0}}}}$, i.e., ${\textstyle \phi _{0}({\vec {x}})=\delta ({\vec {x}}-{\vec {Y_{0}}})}$, the expectation value of ${\textstyle \phi ({\vec {x}},t)}$ is

$${\displaystyle \left\langle \phi ({\vec {x}},t)\right\rangle =f_{X}({\vec {x}};t|Y_{0})}$$

This means that the mean conserved scalar field resulting from a point source is given by the probability density function of the particle position ${\textstyle f_{X}}$ of the fluid particles that originate at the source.

The simplest case to consider is dispersion from a point source, positioned at the origin (${\textstyle Y_{0}=0}$), in statistically stationary isotropic turbulence. In particular, consider an experiment where the isotropic turbulent velocity field has zero mean.

In this setting, one can derive the following results:

• 

  Samples of fluid particle paths given by the Langevin equation for times much shorter than Lagrangian time scale. Note that the expectation evolves linearly. (Both axis are expressed in suitable dimensionless quantities).
  

  Samples of fluid particle paths given by the Langevin equation for times much longer than Lagrangian time scale. Note that the expectation evolves as a square root of time. (Both axis are expressed in suitable dimensionless quantities).
  Given that the isotropic turbulent velocity field has zero mean, fluid particles disperse from the origin isotropically, meaning that mean and covariance of the fluid parcel position are respectively
  $${\displaystyle \left\langle {\vec {X}}(t,0)\right\rangle =\int _{0}^{t}\left\langle {\vec {U}}(s,0)\right\rangle ds=0}$$

  
  $${\displaystyle \left\langle X_{i}(t,0)X_{j}(t,0)\right\rangle =\sigma _{X}^{2}(t)\delta _{ij}}$$

  
  where ${\textstyle \sigma _{x}(t)}$ is the standard deviation and ${\textstyle \delta _{ij}}$ the Kronecker delta.
• The standard deviation of the particle displacement is given in terms of the Lagrangian velocity autocorrelation ${\textstyle \rho (s)}$ following by
  $${\displaystyle \sigma _{X}^{2}(t)=2u'^{2}\int _{0}^{t}(t-s)\rho (s)ds}$$

  
  where ${\textstyle u'}$ is the root mean square velocity. This result corresponds with the result originally obtained by Taylor.[18]
• For all times, the dispersion can be expressed in terms of a diffusivity ${\textstyle {\hat {\Gamma }}_{T}(t)}$ as

$${\displaystyle {\hat {\Gamma }}_{T}(t)={\frac {1}{2}}{\frac {d}{dt}}\sigma _{X}^{2}=u'^{2}\int _{0}^{t}\rho (s)ds}$$

• The quantity ${\displaystyle T_{L}=\int _{0}^{\infty }\rho (s)ds}$ defines a time-scale characteristic of the turbulence called the Lagrangian integral time scale.
• For small enough times (${\textstyle t\ll T_{L}}$), so that ${\textstyle \rho (s)}$ can be approximated with ${\textstyle \rho (0)=1}$, straight-line fluid motion leads to a linear increase of the standard deviation ${\textstyle \sigma _{X}\approx u't}$ which, in term, corresponds to a time-dependent diffusivity ${\textstyle {\hat {\Gamma }}_{T}(t)\approx u'^{2}t}$. This sheds light onto one of the above stated experimental counterexamples to gradient diffusion, namely the observation of linear spreading rate for smoke near chimney.
• For large enough times (${\textstyle t\gg T_{L}}$), the dispersion corresponds to diffusion with a constant diffusivity ${\textstyle \Gamma _{T}=u'^{2}T_{L}}$ so that the standard deviation increases as the square root of time following
  $${\displaystyle \sigma _{X}(t)\approx {\sqrt {2u'^{2}T_{L}t}}}$$

  
• This is the same type of dependence as was derived for a simple case of gradient diffusion. This agreement between the two approaches suggests that for large enough times, the gradient model is working well and instead fails to predict the behavior of particles recently ejected from their source.

The simplest stochastic Lagrangian model is the Langevin equation, which provides a model for the velocity following the fluid particle. In particular, the Langevin equation for the fluid-particle velocity yields a complete prediction for turbulent dispersion. According to the equation, the Lagrangian velocity autocorrelation function is the exponential ${\displaystyle \rho (s)=\exp(-|s|/T_{L})}$. With this expression for ${\displaystyle \rho (s)}$, the standard deviation of the particle displacement can be integrated to yield

$${\displaystyle \sigma _{X}^{2}(t)=2u^{2}T_{L}[t-T_{L}(1-\exp(-t/T_{L}))]}$$

According to the Langevin equation, each component of the fiuid particle velocity is an Ornstein-Uhlenbeck process. It follows that the fluid particle position (ie., the integral of the Ornstein-Uhlenbeck process) is also a Gaussian process. Thus, the mean scalar field predicted by the Langevin equation is the Gaussian distribution

$${\displaystyle \left\langle \phi ({\vec {x}},t)\right\rangle =(\sigma _{X}{\sqrt {2\pi }})^{-3}\exp(-x_{i}x_{i}/2\sigma _{X}^{2})}$$

with ${\displaystyle \sigma _{X}(t)}$ given by the previous equation.

Molecular diffusion is negligible for the purposes of material transport across ocean basins. However, observations indicate that the oceans are under constant mixing. This is enabled by ocean eddies that range from Kolmogorov microscales to gyres spanning entire basins. Eddy activity that enables this mixing continuously dissipates energy, which it lost to smallest scales of motion. This is balanced mainly by tides and wind stress, which act as energy sources that continuously compensate for the dissipated energy.[20][21]

Apart from the layers in immediate vicinity of the surface most of the bulk of the ocean is stably stratified. In a few narrow, sporadic regions at high latitudes surface water becomes unstable enough to sink deeply and constitute the deep, southward branch of the overturning circulation [20] (see e.g. AMOC). Eddy diffusion, mainly in the Antarctic Circumpolar Current, then enables the return upward flow of these water masses. Upwelling has also a coastal component owing to the Ekman transport, but Antarctic Circumpolar Current is considered to be the dominant source of upwelling, responsible for roughly 80% of its overall intensity.[22] Hence the efficiency of turbulent mixing in sub-Antarctic regions is the key element which sets the rate of the overturning circulation, and thus the transport of heat and salt across the global ocean.

Eddy diffusion also controls the upwelling of atmospheric carbon dissolved in upper ocean thousands of years prior, and thus plays an important role in Earth's climate system.[9] In the context of global warming caused by increased atmospheric carbon dioxide, upwelling of these ancient (hence less carbon-rich) water masses while simultaneously dissolving and downwelling present carbon-rich air, causes a net accumulation of carbon emissions in the ocean. This in turn moderates the climate change, but causes issues such as ocean acidification.[10]

An example of horizontal transport that has received significant research interest in the 21st century is the transport of floating plastics. Over large distances, the most efficient transport mechanism is the wind-driven circulation. Convergent Ekman transport in subtropical gyres turns these into regions of increased floating plastic concentration (e.g. Great Pacific garbage patch).[23]

In addition to the large-scale (deterministic) circulations, many smaller scale processes blur the overall picture of plastic transport. Sub-grid turbulent diffusion adds a stochastic nature to the movement. Numerical studies are often done involving large ensemble of floating particles to overcome this inherent stochasticity.

In addition, there are also more macroscopic eddies that are resolved in simulations and are better understood. For example, mesoscale eddies play an important role. Mesoscale eddies are slowly rotating vortices with diameters of hundreds of kilometers, characterized by Rossby numbers much smaller than unity. Anticyclonic eddies (counterclockwise in the Northern hemisphere) have an inward surface radial flow component, that causes net accumulation of floating particles in their centre. Mesoscale eddies are not only able to hold debris, but to also transport it across large distances owing to their westward drift. This has been shown for surface drifters, radioactive isotope markers,[24] plankton, jellyfish,[25][12] heat and salt.[11] Sub-mesoscale vortices and ocean fronts are also important, but they are typically unresolved in numerical models, and contribute to the above-mentioned stochastic component of the transport.[23]

The problem of diffusion in the atmosphere is often reduced to that of solving the original gradient based diffusion equation under the appropriate boundary conditions. This theory is often called the K theory, where the name comes from the diffusivity coefficient K introduced in the gradient based theory.

If K is considered to be constant, for example, it can be thought of as measuring the flux of a passive scalar quantity ${\textstyle \phi }$, such as smoke through the atmosphere.

For a stationary medium ${\textstyle \phi }$, in which the diffusion coefficients, which are not necessarily equal, can vary with the three spatial coordinates, the more general gradient based diffusion equation states,

$${\displaystyle {\frac {\partial \phi }{\partial t}}={\frac {\partial }{\partial x}}\left(K_{x}{\frac {\partial \phi }{\partial x}}\right)+{\frac {\partial }{\partial y}}\left(K_{y}{\frac {\partial \phi }{\partial y}}\right)+{\frac {\partial }{\partial z}}\left(K_{z}{\frac {\partial \phi }{\partial z}}\right)}$$

Considering a point source, the boundary conditions are

$${\displaystyle {\begin{aligned}(1)\quad &\phi \rightarrow 0\quad {\text{as}}\quad t\rightarrow \infty \quad {\text{for}}\quad -\infty <x<\infty \\(2)\quad &\phi \rightarrow 0\quad {\text{as}}\quad t\rightarrow 0\quad {\text{for}}\quad x\neq 0\end{aligned}}}$$

where ${\textstyle \phi \rightarrow \infty }$ such that ${\textstyle \int _{-\infty }^{\infty }\phi dx=\Phi }$, where ${\textstyle \Phi }$ is the source strength (total amount of ${\displaystyle \phi }$ released). The solution of this problem is a Gaussian function. In particular, the solution for an instantaneous point source of ${\textstyle \phi }$, with strength ${\textstyle \Phi }$, of an atmosphere in which ${\textstyle {\overline {u}}}$ is constant, ${\textstyle v=w=0}$ and for which we consider a Lagrangian system of reference that moves with the mean wind ${\textstyle {\overline {u}}}$:

$${\displaystyle {\frac {\phi }{\Phi }}={\frac {1}{(4\pi Kt)^{1/2}}}\exp \left(-{\frac {x^{2}}{4Kt}}\right)}$$

Integration of this instantaneous-point-source solution with respect to space yields equations for instantaneous volume sources (bomb bursts, for example). Integration of the instantaneous-point source equation with respect to time gives the continuous-point-source solutions.

K theory has been applied when studying the dynamics of a scalar quantity ${\displaystyle \phi }$ through the atmospheric boundary layer. The assumption of constant eddy diffusivity can rarely be applied here and for this reason it's not possible to simply apply K theory as previously introduced.

Without losing generality, consider a steady state, i.e. ${\textstyle \partial \phi /\partial t=0}$, and an infinite crosswind line source, for which, at ${\textstyle z=0}$

$${\displaystyle {\frac {\partial }{\partial y}}\left(K_{y}{\frac {\partial \phi }{\partial y}}\right)=0}$$

Assuming that ${\textstyle \partial (K_{x}\partial \phi /\partial x)/\partial x\ll {\overline {u}}\partial \phi /\partial x}$, i.e., the x-transport by mean flow greatly outweighs the eddy flux in that direction, the gradient based diffusion equation for the flux of a stationary medium ${\textstyle q}$ becomes

$${\displaystyle {\overline {u}}{\frac {\partial \phi }{\partial x}}={\frac {\partial }{\partial z}}\left(K_{z}{\frac {\partial \phi }{\partial z}}\right)}$$

This equation, together with the following boundary conditions

$${\displaystyle {\begin{aligned}(1)\quad &\phi \rightarrow 0\quad {\text{as}}\quad z\rightarrow \infty \\(2)\quad &\phi \rightarrow 0\quad {\text{as}}\quad x\rightarrow 0\quad {\text{for all}}\quad z>0\quad {\text{but}}\quad \phi \rightarrow \infty \quad {\text{as}}\quad x\rightarrow 0,\quad z\rightarrow 0\quad {\text{such that}}\quad \lim _{x\rightarrow 0}\int _{0}^{\infty }{\overline {u}}\phi dz=\Phi \\(3)\quad &K_{z}{\frac {\partial \phi }{\partial z}}\quad {\text{as}}\quad z\rightarrow 0\quad {\text{for all}}\quad x>0\end{aligned}}}$$

where, in particular, the last condition implies zero flux at the ground. This equation has been the basis for many investigations. Different assumptions on the form of ${\textstyle K_{z}}$ yield different solutions.

Example of a bent-over plume described using K theory in "Diffusion of stack gasses in very stable atmosphere" by Morton L. Barad.[26]

As an example, K theory is widely used in atmospheric turbulent diffusion (heat conduction from the earth's surface, momentum distribution) because the fundamental differential equation involved can be considerably simplified by eliminating one or more of the space coordinates.[27] Having said that, in planetary-boundary-layer heat conduction, the source is a sinusoidal time function and so the mathematical complexity of some of these solutions is considerable.

In general, K theory comes with some shortcomings. Calder[28] studied the applicability of the diffusion equation to the atmospheric case and concluded that the standard K theory form cannot be generally valid. Monin[29] refers to K theory as a semi-empirical theory of diffusion and points out that the basic nature of K theory must be kept in mind as the chain of deductions from the original equation grows longer and more involved.

That being said, K theory provides many useful, practical results. One of them is the study by Barad[26] where he a K theory of the complicated problem of diffusion of a bent-over stack plume in very stable atmospheres.