Scalar transport

Scalar transport follows a material quantity through advection and diffusion equation terms, for example ${\Theta }_t+\mathbf{U}\cdot \nabla \Theta =\kappa \Delta \Theta$. A passive scalar does not affect the velocity field; an active scalar feeds back through forces such as buoyancy. With incompressible flow bounded by impermeable walls and homogeneous Neumann boundary conditions, the spatial mean is conserved.

An active scalar is a transported scalar field that feeds back on the velocity field, for example a density anomaly producing buoyancy. In an adjoint equations for Boussinesq scalar mixing calculation its dynamical coupling must also be transposed; a passive-scalar sensitivity equation generally omits that feedback.

A passive scalar is an advected and diffusing scalar field whose value has no dynamical effect on the transporting velocity field. Concentration of a dilute nonreacting dye is a common example. Its transport can depend on the flow while the flow equations remain independent of it.

For $\chi =f\exp (-{\mathbf{x}}^{T}B\mathbf{x})$ in two dimensions, take $B$ positive definite, $\mathrm{tr}C=0$ and positive conserved mass $Q=\int \chi dA$. The Gaussian integral gives $Q=\pi f/\sqrt{\det B}$, $J=(Q/2)B^{-1}$ and $f=Q^2/(2\pi \sqrt{\det J})$. Substitution in the advection-diffusion equation gives $\dot{B}=-C^{T}B-BC-4\kappa B^2$ and $\dot{f}/f=-2\kappa \mathrm{tr}B$, equivalently the displayed moment equation. If $\dot{F}=CF$, its solution is $J=F[J_0+2\kappa Q{\int }_0^t{F}^{-1}(s)F^{-T}(s)ds]F^T$, with $\det F=1$.

For a chaotic incompressible planar flow with Lyapunov exponent pair $\lambda ,-\lambda$, an initially broad Gaussian scalar packet in a linear incompressible flow stretches and compresses exponentially while diffusion is negligible. Its compressed variance stops decreasing around $\kappa /\lambda$, the Batchelor scalar microscale balance, while the larger variance continues to grow. In a constant principal strain, the normalized covariance eigenvalues obey ${\dot{j}}_{\pm }=\pm 2\lambda j_{\pm }+2\kappa$, proving that the compressed one approaches $\kappa /\lambda$. Random finite-correlation strain instead gives a fluctuating compressed width; stationarity and integrability assumptions are required to turn this into exponential statistics of the packet amplitude.

After diffusion arrest of Gaussian packet compression, define the logarithmic area increase $A_t=\frac{1}{2}\log [\det J(t)/\det J(0)]$. It is nonnegative. Assume $A_t/t$ has a large deviation principle with convex rate function $\mathcal{I}$, minimized at $\lambda >0$. The exact amplitude relation $f(t)=f(0)e^{-A_t}$ and the Laplace principle for probability measures give the displayed moment decay rate. The rate function refers to area, so rare width fluctuations are included rather than discarded without justification. For differentiable strictly convex $\mathcal{I}$, the interior minimizer satisfies ${\mathcal{I}}^{\prime }(s_{\mu })=-\mu$. If ${\mu }_{\ast }=-{\mathcal{I}}^{\prime }(0)$ is finite, then ${\gamma }_{\mu }=\mathcal{I}(0)$ for $\mu \ge {\mu }_{\ast }$: rare nearly unstretched packets dominate high moments. Below the threshold, $d({\gamma }_{\mu }/\mu )/d\mu =-\mathcal{I}(s_{\mu })/{\mu }^2<0$. These statements concern a positive Gaussian packet with flow-only averaging; a homogeneous signed random initial scalar requires a different averaging kernel. Arbitrary random $C(t)$ alone does not imply this statistical model.

The molecular Schmidt number is the ratio of kinematic viscosity to scalar diffusivity. A large value allows a passive scalar to develop structure below the viscous smoothing scale of the velocity field. It is distinct from the turbulent Schmidt number, which compares effective turbulent transport coefficients, and from the Schmidt number used for quantum entanglement.

Differentiate the advection equation componentwise: $D_t({\partial }_i\theta )=-({\partial }_i{u}_j){\partial }_j\theta$. Along a Lagrangian trajectory, the scalar gradient therefore evolves by the negative transpose of the velocity gradient tensor. If $F$ is the deformation gradient, this says $\mathbf{k}(t)=F(t{)}^{-T}\mathbf{k}(0)$, as expected for a covector. Fluid material vectors and scalar gradients transform oppositely under stretching.

For horizontal strain matrix $S=\left(\begin{array}{cc}a& b\\ b& -a\end{array}\right)$ and zero vertical velocity, horizontal scalar gradients obey ${\dot{k}}_h=-S{k}_h$. Since $S^2=c^{2}I$ with $c=\sqrt{a^2+b^2}$, the growing projection is $(I-S/c)k_h(0)/2$. If nonzero it equals $k_0(\cos \varphi ,\sin \varphi )$ and grows as $e^{ct}$. Fixed vertical shears give ${\dot{k}}_3=-{\sigma }_{13}{k}_1-{\sigma }_{23}{k}_2$, so $k_3\sim -k_{0}D{e}^{ct}/c$ with $D={\sigma }_{13}\cos \varphi +{\sigma }_{23}\sin \varphi$. The ratio $\alpha =|k_3|/|k_h|$ tends to the displayed value. It measures scalar-isosurface orientation; zero growing projection or $D=0$ must not be described as generic exponential vertical-gradient growth.

In the long-time growing horizontal-gradient branch, independent vertical shear noises combine as $\cos \varphi d{W}_1+\sin \varphi d{W}_2=dW$. The signed inclination $\beta =k_3{e}^{-ct}/k_0$ satisfies the displayed Ornstein-Uhlenbeck stochastic differential equation. For $c>0$ its stationary density is $\sqrt{c/(\pi g^2)}{e}^{-c{B}^2/g^2}$, with variance $g^2/(2c)$. The Fokker-Planck probability current vanishes at infinity and gives this Gaussian by one integration. At $c=0$ and nonzero noise there is no normalizable stationary density.

Let $c=\sqrt{a^2+b^2}$ with independent $a,b\sim N(0,{\Gamma }^2)$, and let $D\sim N(0,{\Lambda }^2)$ be independent of them. Then $c$ has Rayleigh distribution and $\alpha =|D|/c$ has the displayed cumulative distribution function for $A\ge 0$ and density ${\Lambda }^2\Gamma ({\Gamma }^2{A}^2+{\Lambda }^2{)}^{-3/2}$. Indeed, conditioning on $D$ gives $\mathbb{P}(c\ge |D|/A)=\exp [-D^2/(2{\Gamma }^2{A}^2)]$, and a Gaussian integral completes the calculation. Independence is essential: choosing $D=(\Lambda /\Gamma )a$ has the same Gaussian marginal but makes the ratio bounded by $\Lambda /\Gamma$.

For a homogeneous scalar, $S_C(r)=2[⟨C^2⟩-⟨C{C}^{\prime }⟩]$. Decorrelation at large separation gives saturation at twice the variance. If the scalar is mean-square differentiable and isotropic, $S_C(r)=r^2⟨|\nabla C{|}^2⟩/3+o(r^2)$ at small separation. Statistical isotropy and zero mean alone do not imply decorrelation at infinity.

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.

For scalar diffusivity much larger than kinematic viscosity, scalar diffusion dominates at lengths between the Kolmogorov and scalar cutoff scales. The velocity may still have inertial-range variations, but the scalar is smooth to leading order. The leading isotropic scalar structure function is quadratic, with coefficient fixed by the mean-square gradient and the scalar dissipation rate, rather than by a continuing inertial-convective two-thirds law.

In the large-diffusivity regime, equate diffusion rate $\alpha /r^2$ with inertial eddy rate ${ϵ}^{1/3}{r}^{-2/3}$. This gives the Corrsin scalar microscale, which exceeds the Kolmogorov length scale when $\alpha \gg \nu$.

For scalar diffusivity much smaller than kinematic viscosity, a scalar has structure below the Kolmogorov length scale while the velocity is smooth. The strain rate $s\sim (ϵ/\nu {)}^{1/2}$ transfers scalar variance, giving $d{S}_C/dr\sim {ϵ}_c/(sr)$. Integration produces a logarithm plus an additive matching constant.

At the Batchelor scalar microscale, molecular diffusion rate $\alpha /r^2$ balances the smooth Kolmogorov strain rate $(ϵ/\nu {)}^{1/2}$. For $\alpha \ll \nu$ this scale is much smaller than the Kolmogorov length scale.

This range lies above both the Kolmogorov length scale and the scalar diffusion cutoff, but below the forcing scale. Molecular viscosity and scalar diffusion have negligible direct influence on transfer at those scales. The Obukhov-Corrsin theory predicts a two-thirds scalar structure function there.

The horizontally integrated plume concentration is the integral of a local concentration $C$ across the plume. It measures scalar amount per unit vertical length and unit span, and its vertical integral is total scalar amount per unit span. Unlike a local concentration, its dimensions include one horizontal length. It must be distinguished from horizontally averaged plume concentration when a plume's width varies with height.

For a plume of full width $w(z)$, its horizontally averaged concentration is $\overline{C}(z,t)=c(z,t)/w(z)$, where $c$ is horizontally integrated plume concentration. In a line plume with $w=\mu z$, division by $z$ generally changes the location of maximum concentration. A gradient closure for $c$ differs from one for $\overline{C}$: $w{\partial }_z\overline{C}=c_z-c/z$, so integrated diffusive flux $-Kw{\partial }_z\overline{C}$ equals $-K(c_z-c/z)$.

For a smooth incompressible direct trajectory with total velocity field $\mathbf{U}$ and active scalar $\Theta$, the negative-constraint Lagrangian function in constrained optimization convention givesThe transpose term is the formal adjoint of $\delta \mathbf{u}\cdot \nabla \mathbf{U}$; the two couplings transpose scalar transport by advection and buoyancy. For terminal cost $\int \Theta (T{)}^2$, the terminal data are ${\mathbf{u}}^{†}(T)=0$, ${\theta }^{†}(T)=2\Theta (T)$. They are integrated backward along the stored direct trajectory. No-slip boundary conditions and Dirichlet boundary conditions for the adjoint velocity and homogeneous Neumann boundary conditions for the adjoint scalar remove the spatial boundary terms.

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.

With this half-variance convention, the unforced scalar equation gives $d⟨C^2/2⟩/dt=-{ϵ}_c$ in a homogeneous incompressible flow. Authors defining dissipation of the full variance instead use $2\alpha ⟨|\nabla C{|}^2⟩$. Stating the convention is necessary when relating the dissipation rate to a scalar structure function.