Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 87 / 2

The three questions concern different statistics. Taylor turbulent dispersion follows the absolute displacement of one marked fluid element from its release position, or the spreading of the centre of a cloud. Its input is the velocity autocorrelation along a Lagrangian trajectory. It relates short-time ballistic motion to long-time effective diffusion. This is not the shear-flow mechanism ordinarily called Taylor dispersion.

Richardson pair dispersion instead follows the separation of two fluid elements, or the growth of a cloud relative to its moving centre. Common sweeping by an eddy much larger than the cloud contributes to absolute displacement but largely cancels from relative displacement. The relative diffusivity depends on separation; the classical inertial-range prediction is a $t^3$ mean-square separation regime after memory of the release separation is lost.

Obukhov-Corrsin theory, also called Kolmogorov-Obukhov-Corrsin theory, concerns spatial fluctuations of a passive scalar, their transfer across scales and their destruction by molecular diffusion. Its objects are scalar structure functions and scalar spectra, with an inertial-convective prediction controlled by scalar-variance flux and the turbulent energy supply. Thus Taylor describes one-particle transport, Richardson describes relative spreading, and Obukhov-Corrsin describes scale-by-scale scalar mixing. Following a cloud's location is not the same as predicting its internal concentration fluctuations.

Since the particle starts at the origin, its Lagrangian trajectory satisfies $\mathbf{X}(t)={\int }_0^t\mathbf{v}(s)ds$. Differentiating the mean-square displacement givesHere we set $\tau =t-s$ and assume finite second moments so that averaging and integration commute. Under Lagrangian stationarity write the autocorrelation as $R_L(\tau )$. Uniform fluid-tracer sampling in homogeneous incompressible turbulence gives $R_L(0)=⟨|\mathbf{u}{|}^2⟩$. A second integration yields the exact Taylor turbulent dispersion formulaFor times short compared with velocity decorrelation, continuity at zero gives $R_L(\tau )\simeq R_L(0)$ throughout the integral. ThusThe particle retains nearly its initial velocity: this is ballistic transport.

If $R_L$ is integrable, then at long times the displacement derivative approaches $2{\int }_0^{\infty }{R}_L(\tau )d\tau =2R_L(0)t_L$, using the Lagrangian integral time. ThereforeThis is diffusive transport: successive long-time displacements lose memory of one another. In an isotropic three-dimensional diffusion equation, $⟨X^2⟩=6D_{T}t$, so $D_T=⟨|\mathbf{u}{|}^2⟩t_L/3$. The shorthand $t\ll t_L$ and $t\gg t_L$ presumes that $t_L$ is representative of the decorrelation time. With strongly oscillatory correlations, the area-based integral time need not characterize every crossover; the exact integral formula remains the appropriate starting point.

Measure the cloud radius about its moving centre, so uniform sweeping does not count as spreading. In the inertial range, an eddy of size $R$ has velocity difference $u_R\sim (ϵR{)}^{1/3}$ and turnover time $t_R\sim R/u_R$. A scale-local relative diffusivity is thereforegiving the estimateIt requires separation from the molecular and viscous cutoffs and from the integral scale of turbulence, high Reynolds number, approximately steady energy supply, and scale-local relative motion. Mean shear, walls, particle inertia or coherent large-scale strain can invalidate this simple diffusivity model. At very short times a pair retains its initial relative velocity, so a diffusion closure is not yet appropriate.

Let $s(t)=⟨|\delta \mathbf{x}(t){|}^2⟩$. A self-similar pair-separation distribution, together with the relative-diffusivity estimate, givesSelf-similarity is needed to replace a separation-dependent moment such as $⟨r^{4/3}⟩$ by a fixed multiple of $s^{2/3}$; stationarity alone does not do that. Integrating this model givesAfter loss of initial-separation memory, it reduces to the Richardson pair dispersion lawThe required time window is $t\gg t_0\sim (r_0^2/ϵ{)}^{1/3}$ while the separation remains far below the integral scale. The stated stationarity and inertial-range separation conditions alone cannot imply the displayed $t^3$ law at every time: a nonzero release separation already gives $s(0)=r_0^2$, whereas $gϵt^3$ vanishes at zero. At sufficiently short times, $\delta \mathbf{x}(t)-\delta {\mathbf{x}}_0\simeq \delta {\mathbf{u}}_{0}t$, so its mean-square change is ballistic and remembers $r_0$.

The Richardson constant might be universal if the inertial-range relative dynamics, including the normalized separation distribution, become independent of release details and forcing at sufficiently high Reynolds number. This is an additional universality hypothesis. As evidence available before 2008, Boffetta and Sokolov's numerical study estimated a coefficient near $0.55$ and reported agreement with experimental estimates, alongside small deviations from ideal Richardson scaling. Bourgoin and collaborators' 2006 experiment showed substantial dependence on initial separation in the accessible laboratory regime. Values of order $0.5$ are supported in suitable regimes, but finite-time fits do not establish one universally observed $t^3$ coefficient for all releases.

Use $S_C(r)=⟨(C^{\prime }-C{)}^2⟩$ for the scalar structure function, and assume statistical homogeneity as well as isotropy. Expanding the square givesIf the zero-mean scalar decorrelates at large separation, thenThe decorrelation assumption is necessary: a spatially constant, zero-ensemble-mean random scalar is isotropic but has $S_C=0$ at every separation.

For the small-separation limit assume mean-square differentiability. Taylor expansion gives $C^{\prime }-C=r_i{\partial }_{i}C+o(r)$ in mean square. Isotropy implies $⟨{\partial }_{i}C{\partial }_{j}C⟩={\delta }_{ij}⟨|\nabla C{|}^2⟩/3$. Define the scalar dissipation rate in the half-variance convention, ${ϵ}_c=\alpha ⟨|\nabla C{|}^2⟩$, so that diffusion removes $⟨C^2/2⟩$ at rate ${ϵ}_c$. It follows thatIf dissipation is instead defined as the loss rate of the full variance, the coefficient must be adjusted by a factor of two.

In the inertial-convective scalar range, $\max (\eta ,{\eta }_c)\ll r\ll \ell$, suppose scalar variance cascades locally at an approximately constant flux ${ϵ}_c$, while velocity increments have the Kolmogorov 1941 theory turnover time $t_r\sim {ϵ}^{-1/3}{r}^{2/3}$. The variance associated with scale $r$ divided by that turnover time is of order its transfer rate, giving the Obukhov-Corrsin theory resultThe assumptions include a passive scalar, high Reynolds number, local transfer, adequate scale separation, and statistical equilibrium or sufficiently slow variation of the supply. Isotropy alone does not provide a constant scalar flux.

For weak scalar diffusion, $\alpha \ll \nu$, the velocity is smooth below $\eta$, with characteristic strain rate $s\sim (ϵ/\nu {)}^{1/2}$. Diffusion rate $\alpha /r^2$ balances this at the Batchelor scalar microscaleIn the viscous-convective scalar range, the derivative of $S_C$ is assumed to depend only on ${ϵ}_c$, $s$ and $r$. Their dimensions forceIntegration from the diffusion cutoff givesThus, well within a broad logarithmic range,The additive matching contribution is not exactly zero; the formula is a leading range estimate, not a boundary condition at $r={\eta }_c$. Using $d{S}_C/dr$ isolates the contribution acquired on changing separation. The structure function itself contains scalar variance accumulated from smaller scales and must retain the inner cutoff through its integration constant. If one assumed that $S_C$ depended only on ${ϵ}_c,s,r$ without that cutoff, dimensional analysis would incorrectly give an $r$-independent value proportional to ${ϵ}_c/s$. This explains the use of the derivative with respect to $r$; the TeX aid changes that derivative to $t$ in its final sentence, but the original PDF consistently uses $r$.

Finally let $\alpha \gg \nu$. Balancing diffusion with the inertial eddy rate gives the Corrsin scalar microscale ${\eta }_c={\eta }_C\sim ({\alpha }^3/ϵ{)}^{1/4}\gg \eta$. For $\eta \ll r\ll {\eta }_c$, diffusion is faster than the eddy turnover rate. The scalar is smooth to leading order on those separations, even though the velocity still has inertial-range fluctuations. The inertial-diffusive scalar range therefore hasThis continues the scalar's leading gradient-controlled quadratic behaviour; it is not a continuation of the inertial-convective two-thirds law below its diffusion cutoff.