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Obukhov-Corrsin theory concerns transfer of scalar variance across scales in turbulent mixing. In the inertial-convective range, a constant scalar-variance flux and an eddy turnover time ${ϵ}^{-1/3}{r}^{2/3}$ predict the displayed second-order scalar structure function. It assumes a passive scalar, local transfer, approximate statistical equilibrium and separation from the molecular diffusion and forcing scales.

A passive scalar $\theta$ is transported by the velocity field without feeding back on it:Take its mean to be zero, and define the scalar dissipation rate by the half-scalar variance convention $\chi =\kappa ⟨|\nabla \theta {|}^2⟩$. In a statistically equilibrated energy cascade of scalar fluctuations, this is also the flux of half-scalar variance toward small scales.

The Kolmogorov two-thirds law gives a typical velocity increment $\delta u_r\sim (ϵr{)}^{1/3}$ and hence eddy turnover time ${\tau }_r\sim r/\delta u_r\sim {ϵ}^{-1/3}{r}^{2/3}$. Assuming local scalar transfer on this same time scale, $\chi \sim ⟨(\Delta \theta {)}^2⟩/{\tau }_r$. The Obukhov-Corrsin theory therefore gives the scalar analogue:This applies to separations at which both direct forcing and molecular diffusion are negligible and the transporting velocity increments lie in the inertial range. For very large or very small ratios $\nu /\kappa$, the scalar and velocity cutoff scales differ, so this common range must be checked. Defining $\chi$ as dissipation of the full scalar variance instead simply changes the convention for $C_{\theta }$.

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.

Use $\nu =R{e}^{-1}>0$, $\kappa =P{e}^{-1}>0$, $\mathbf{U}=\overline{\mathbf{u}}+\mathbf{u}$, and $\Theta =\overline{\theta }+\theta$. Work along a sufficiently regular direct trajectory satisfying the given constraints. The zero spatial mean of the scalar field is conserved by incompressible flow, impermeable walls and zero scalar flux, so minimizing $J=(\Theta (T),\Theta (T))$ is equivalent to minimizing scalar variance, up to the fixed domain volume.

The printed functional fixes the initial state to a candidate ${\mathbf{u}}_0$; it contains no term that enforces its kinetic energy. For the optimization over that candidate, add the real Lagrange multiplier constraintEquivalently, one can restrict all control variations to the sphere in a normed vector space of fixed kinetic energy. This term changes the initial-control optimality condition, not the interior adjoint equations.

Let $\mathbf{v}=\delta \mathbf{u}$, $\sigma =\delta \theta$ and $\pi =\delta p$. Linearization of the momentum and scalar transport residuals givesBoth appearances of the perturbation velocity in the nonlinear momentum term have been differentiated. In particular, the coefficient is the gradient of the total velocity, not just the base shear.

For the negative-constraint convention of the functional, integration by parts gives the interior coefficients of $\mathbf{v},\sigma ,\pi$ asThus the adjoint equations for Boussinesq scalar mixing areThe transpose is essential: the $j$th component of $(\nabla \mathbf{U}{)}^T{\mathbf{u}}^{†}$ is ${\sum }_i{u}_i^{†}{\partial }_j{U}_i$. The coupling $-{\theta }^{†}\nabla \Theta$ transposes advection of the scalar by a velocity perturbation; $-R{i}_B{u}_y^{†}$ transposes buoyancy feedback. Dropping the latter would give a passive scalar adjoint, not the active scalar problem.

These equations are integrated backward, not forward. If $\tau =T-t$, they readwith direct coefficients evaluated at $t=T-\tau$. Both terms from the diffusion equation now have the usual forward sign in $\tau$. A direct-adjoint looping method stores or reconstructs the forward trajectory, solves these equations backward, and uses the initial adjoint as the control gradient. The endpoint and fixed-energy conditions below give necessary conditions for a local optimizer, not a global optimality theorem.

Scalar variance measures spatial departure from the conserved mean. For zero mean, incompressible flow bounded by impermeable walls and homogeneous scalar Neumann boundary conditions, integration by parts gives $d\int {\Theta }^2/dt=-2\kappa \int |\nabla \Theta {|}^2$. Advection preserves the instantaneous quadratic integral but can sharpen gradients, allowing the diffusion equation to remove scalar variance faster.