4: Turbulent Diffusion
- Page ID
- 1273
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)On the scale of the ocean, diffusion due to the random thermal motion of molecules is very slow. With a diffusion coefficient of \(\sim 2 * 10^{-9}\) m\(^2\)/s, diffusive transport over a distance of just 100 m takes about 100000 years. However, transport rates in both the ocean and the atmosphere are enhanced by many orders of magnitude through stirring by eddies of various sizes that are collectively named 'turbulence'. The study of turbulence forms a very large and complicated topic that we are not going to cover in-depth here. Instead, we will try to provide an idea of turbulence as a phenomenon and of the role it plays in the ocean and the atmosphere.
Turbulence can have a mechanical or a convective origin. As an everyday example of mechanical production of turbulence, take a look at this instructional video by Tony Kuphaldt of Bellingham Technical College.
As the tap is opened further and further and the velocity of the water increases, the flow changes from laminar (regular and transparent) to turbulent which means that it is full of little eddies. At the ocean surface, vigorous mechanical turbulence production can take place when a storm blows over the water. Turbulence is produced convectively, when a fluid is heated from below or cooled from above. The cooler, denser fluid on top tends to sink, while the warmer, lighter fluid at the bottom tends to rise, leading to the formation of eddies. When the ocean surface cools during winter (particularly at high latitudes), the result is convective turbulence production. However, in most places, the ocean is stably stratified most of the time, that is, the ocean surface is warmer than the deeper water. This tends to dampen true three-dimensional turbulence which is why oceanic turbulence is to a large extent two-dimensional. Therefore, the horizontal turbulent diffusivity (\(K_h\)) is typically orders of magnitude larger than the vertical turbulent diffusivity (\(K_v\)). Turbulence, whether two- or three-dimensional, can lead to rapid and erratic dispersion. Overall, the effective dispersion of the momentum of the large-scale background flow over rather large distances is the main reason why turbulence is important for the large-scale ocean circulation. To a satisfactory approximation, this can be represented as a diffusion process with different horizontal and vertical diffusion coefficients:
\[\dfrac{dv}{dt}=K_h\left(\dfrac{d^2v}{dx^2}+\dfrac{d^2v}{dy^2}\right)+K_v\dfrac{d^2v}{dz^2}\]