Turbulence

Turbulence is fluid flow with irregular, interacting velocity fluctuations over a range of lengths and times. An unsteady laminar flow also has a time-dependent velocity field; the important distinction is the organization and interaction of the fluctuations. In many flows a large Reynolds number allows nonlinear advection to distort the motion before kinematic viscosity smooths it, but transition depends on geometry and disturbances as well as the chosen Reynolds number.

The Reynolds pipe-and-dye experiment makes the distinction visible. A narrow band of colored water can remain coherent in a glass pipe, then break into irregular structures and spread across the pipe as the discharge increases. The inlet and pre-existing disturbances matter, so the observation does not establish a universal transition value.^[ref]

In a three-dimensional forward energy cascade, nonlinear interactions transfer kinetic energy towards smaller scales until viscous dissipation converts it into heat. This transfer helps explain why a turbulent flow mixes a passive scalar efficiently, although final molecular homogenization still requires molecular diffusion. After Reynolds averaging, correlations of fluctuating velocities produce the Reynolds stress, an additional momentum flux. Predicting that flux from mean quantities is the turbulence closure problem; a model for it is an extra physical assumption.

Turbulence intensity expresses the size of a measured velocity fluctuation relative to a specified mean speed. For a streamwise velocity record $u(t)$, defineHere the symbol in the square-root definition is $u_{\mathrm{rms}}$, the root mean square fluctuation, and angle brackets denote the averaging operation. The dimensionless intensity is often reported as $100I$ percent. The averaging interval and measured velocity component are part of the definition.

For four equally weighted observations $9,11,9,11\mathrm{m}{\mathrm{s}}^{-1}$, the mean is $10\mathrm{m}{\mathrm{s}}^{-1}$ and each fluctuation is $\pm 1\mathrm{m}{\mathrm{s}}^{-1}$. Their record root mean square is therefore $1\mathrm{m}{\mathrm{s}}^{-1}$, giving $I=0.10$, or 10 percent. This calculation describes that record; it does not estimate every fluctuation timescale from four observations.

A primary wind-tunnel investigation used hot-wire anemometry, recording mean and fluctuating signals to characterize the wind-tunnel turbulence intensity. Its bandwidth comparison demonstrates that integrating only part of the measured fluctuation spectrum changes the recovered root mean square.^[ref] An instrument's temporal response, filtering and record duration therefore affect the reported intensity. Slow changes of the mean can also contribute to a measured fluctuation unless the averaging convention separates them.

A three-component convention instead useswhere $k$ is the turbulent kinetic energy per unit mass. This agrees with the streamwise definition when the three fluctuation variances are equal, as in isotropic turbulence. A nonzero root mean square alone does not establish turbulence: a periodic laminar velocity oscillation also has a nonzero value. The ratio also becomes ill-conditioned near zero mean speed.

A large eddy simulation computes the time-dependent large-scale motion of a turbulent flow while representing the influence of unresolved motion through a subgrid-scale model. A spatial averaging filter specifies which information is retained. The distinction concerns spatial resolution, rather than simply replacing a fluctuating flow by its time average.^[ref]

For an elementary illustration, take one component of a periodic velocity field,An ideal filter retaining Fourier modes with integer index at most four keeps the first term and removes the second, whose index is ten. The removed term still contributes to products such as $u_x^2$. Filtering the nonlinear Navier-Stokes equations consequently introduces a subgrid-scale stress that cannot generally be reconstructed from the retained velocity field alone.

This gives a computational compromise. Direct numerical simulation must resolve the dynamically relevant dissipative motion, whereas a large eddy simulation resolves a chosen coarser field and models its missing influence. Doubling grid counts in all three directions multiplies the cell count by eight, before any additional time-step cost. The filter, numerical discretization and closure must work together; a coarse grid by itself does not certify a useful approximation.

An early concrete demonstration was James W. Deardorff's 1970 calculation of turbulent channel flow using 6720 uniform grid intervals and strain-dependent coefficients for subgrid effects. The paper compared computed statistics with experimental measurements and reported agreement ranging from good to marginal. That comparison tests a particular simulation, rather than establishing that every resolved structure or model is exact.^[ref]

Implementations can supply subgrid dissipation in different ways. The Cardinal documentation, for example, describes a spectral filter used to drain energy from the smallest retained wavelengths; that is a particular numerical treatment, not the definition of all large eddy simulation.^[ref]

A spatial averaging filter maps a spatially varying quantity to a weighted local average. For a translation-invariant kernel $G_{\Delta }$ of characteristic width $\Delta$,This is a convolution. The normalization preserves a constant value; the kernel determines how each spatial scale is changed. In large eddy simulation, the filter defines the resolved quantities whose evolution is calculated.^[ref]

For a one-dimensional box averaging filter,Integrating $f(x)=\cos (kx)$ explicitly givesThe multiplier is a sinc function. A wave with $k\Delta =\pi$ retains amplitude $2/\pi$, while one with $k\Delta =2\pi$ is removed. This filter attenuates different Fourier modes differently; it is not an ideal sharp cutoff of all modes above one threshold.

A linear filter does not preserve products: generally $\overline{fg}\ne \bar{f}\bar{g}$. That lost product information produces the subgrid-scale stress. Nor does every filter satisfy $\overline{\bar{f}}=\bar{f}$; applying the box average twice produces a different kernel.

With constant width on an unbounded or periodic region, differentiating the convolution gives ${\partial }_j\bar{f}=\overline{{\partial }_{j}f}$ under suitable regularity. A width varying with position or a special near-wall construction can introduce additional terms. Those assumptions must be stated before commuting a filter through a partial derivative.

A grid filter defines the resolved field associated with a large eddy simulation, commonly written $\bar{u}=G_{\Delta }\ast u$. Its name reflects the connection between resolved scales and a computational grid, but the filtering operator and the mesh are different specifications. An explicit spatial averaging filter states its kernel; in other methods the retained representation and numerical operations supply an implicit scale selection.^[ref]

A characteristic width is often related to cell dimensions. One possible three-dimensional convention isCells with widths $0.01$, $0.02$ and $0.04\mathrm{m}$ then have $\Delta =0.02\mathrm{m}$. This single number preserves the volume scale but does not describe the strong directional differences of the cell. The filter shape and directional resolution still matter.

Sampling alone also cannot determine which unresolved oscillations were present. At eight equally spaced points on a periodic interval of length $L$, the modes with indices ten and two have identical sine samples:This is aliasing. Removing the high mode by a specified filter before sampling and merely sampling the unfiltered field are different operations.

The subgrid-scale stress depends on the filter actually adopted. A test filter then applies an additional operation to the already resolved data. Reporting only a grid spacing, without the filtering and discretization choices, can leave the mathematical meaning of the resolved field and its closure ambiguous.

A box averaging filter gives equal weight to all positions within a specified window and zero weight outside it. In one spatial dimension, its normalized kernel and action areThis is a spatial averaging filter expressed as a convolution. In several dimensions, averaging over an axis-aligned rectangular box uses the product of the one-dimensional kernels.

The discrete counterpart averages a fixed number of samples. On an infinite sequence that is zero except for value ten at index zero, a centred five-sample box average produces value two at indices $-2,-1,0,1,2$ and zero elsewhere. The total remains ten, while the isolated peak spreads. On a finite array, boundary extension must be specified; reflection, zero padding and wrapping can give different edge values.^[ref]

Applying the same box average twice is not the same operation as applying it once. The new kernel is the convolution of the two kernels:Indeed, the integral counts the length of overlap of two intervals, divided by ${\Delta }^2$. The resulting support has width $2\Delta$, and the weight decreases away from its centre.

For a Fourier mode $\cos (kx)$, one application multiplies the amplitude by $\sin (k\Delta /2)/(k\Delta /2)$; two applications square this multiplier. The sinc function response explains why a box average smooths a signal without implementing an ideal sharp frequency cutoff. In large eddy simulation, that distinction matters when relating a grid filter to a coarser test filter.

A subgrid-scale model represents the influence of unresolved motion on the quantities retained in a large eddy simulation. Its need follows directly from the nonlinear velocity product in the Navier-Stokes equations: averaging a product generally differs from multiplying the averages.

Let a bar denote a spatial averaging filter. With constant mass density $\rho$, constant kinematic viscosity $\nu$, and a time-independent translation-invariant filter on an unbounded or periodic domain, filtering the incompressible momentum equation giveswhere repeated indices use the Einstein summation convention, $p$ is pressure, and the subgrid-scale stress in this sign convention isThe assumed filter commutes with the displayed derivatives. The definition of ${\tau }_{ij}$ is exact; a closure must approximate it using available resolved information.^[ref]

For an elementary example, average two equally weighted values of one velocity component, $a$ and $-a$. The mean is zero but the mean square is $a^2$, so the corresponding stress component is $a^2$. A resolved mean alone has lost the information needed to reconstruct that covariance.

The Smagorinsky model closes the deviatoric part with an eddy viscosity:where ${\delta }_{ij}$ is the Kronecker delta, ${\bar{S}}_{ij}=({\partial }_j{\bar{u}}_i+{\partial }_i{\bar{u}}_j)/2$ is the resolved strain-rate tensor, $C_s$ a dimensionless model coefficient and $\Delta$ a characteristic filter width. The isotropic stress can be included in a modified pressure. This strain-dependent viscosity has its historical origin in Joseph Smagorinsky's atmospheric circulation modelling.^[ref][ref]

For an incompressible flow, the model transfers resolved kinetic energy out at the rate per unit massA fixed nonnegative eddy viscosity therefore permits dissipation but excludes local turbulent backscatter. A dynamic subgrid-scale model can instead estimate its coefficient from resolved stresses at two filter levels. This addresses a modelling choice, rather than changing the exact definition of the unresolved stress.^[ref]

A dynamic subgrid-scale model estimates its closure coefficient from the resolved flow as the calculation proceeds. In a common construction, a second, coarser test filter probes interactions among resolved scales, providing information for a subgrid-scale model below the original resolution.^[ref]

Write a bar for the original spatial averaging filter and a hat for the test filter. DefineDirect subtraction gives the Germano identityThe rightmost expression uses only resolved quantities. The identity is exact for these definitions; extrapolating a closure between the filter levels is the modeling step.^[ref]

For instance, suppose a chosen closure writes the deviatoric stresses as ${\tau }^d=CA$ and $T^d=CB$, and assumes $C$ can be taken outside the test filtering. Then $L^d\approx CM$, where $M=B-\hat{A}$. One way to choose $C$ is to minimize ${\sum }_{ij}(L_{ij}^d-C{M}_{ij}{)}^2$. Differentiation giveswhen the denominator is nonzero. This is a least squares fit of a scalar to tensor data. In a one-component illustrative fit, $L^d=0.012$ and $M=0.3$ give $C=0.04$.

The coefficient need not remain positive. Depending on the closure, a negative value can permit turbulent backscatter, but a small denominator can also make an estimate unreliable. Averaging, bounds and treatment of weak strain are additional choices. Dynamically fitting a coefficient does not remove these assumptions or make unresolved motion observable.

A test filter is an additional spatial averaging filter applied to already resolved data in a dynamic subgrid-scale model. If $\bar{f}=G\ast f$ is the field after the grid filter, a test kernel $H$ givesIt probes interactions within the available resolved motion at a coarser analysis level. It does not measure the original subgrid motion directly.^[ref]

For example, suppose two resolved values of one velocity component are $1$ and $3\mathrm{m}{\mathrm{s}}^{-1}$. A test average gives $\hat{\bar{u}}=2\mathrm{m}{\mathrm{s}}^{-1}$ and $\widehat{{\bar{u}}^2}=5{\mathrm{m}}^2{\mathrm{s}}^{-2}$, so the resolved differenceis nonzero. This quantity is computable from the resolved samples; a dynamic subgrid-scale model uses such information to constrain a closure.

The combined filter is $H\ast G$, so its width cannot always be identified with the width of $H$ alone. A centred box averaging filter of width $\Delta$ has second moment ${\Delta }^2/12$. Combining it with one of width $2\Delta$ gives second moment $5{\Delta }^2/12$, because centred convolution kernels add their second moments. A single box of width $\sqrt{5}\Delta$ has the same second moment, but not the same kernel shape.

Choosing a width convention is therefore part of the model. The Germano identity holds for the filters actually defined; an assumed relation between modeled stresses at those levels is an additional approximation. Near boundaries, the test operation also needs a specified treatment of data outside the available region.

The Germano identity is an exact relation between stresses at two filtering levels and a stress computable from the resolved field. Let a bar denote a grid filter and a hat a test filter. With the subgrid-scale stress conventionthe identity statesIt follows by substituting ${\hat{\tau }}_{ij}=\widehat{\overline{u_i{u}_j}}-\widehat{{\bar{u}}_i{\bar{u}}_j}$ and cancelling the common term. Massimo Germano and collaborators used this relation to construct a dynamic subgrid-scale model.^[ref]

A scalar block-average example makes every term visible. Start with four velocity values grouped as $(0,2)$ and $(2,4)$. Averaging within each group gives $\bar{u}=(1,3)$ and $\overline{u^2}=(2,10)$, hence $\tau =(1,1)$. A test average over the two groups then givesThus $L=T-\hat{\tau }$ reads $1=2-1$. The resolved group averages contain enough information for $L$, although they alone do not determine the original within-group variances.

The identity has no fitted coefficient. To estimate a missing subgrid-scale stress, a closure assumes some model form at both filtering levels and tries to make their difference agree with the known $L_{ij}$. Extracting a varying coefficient from a filter, choosing a filter width and stabilizing a fit are modeling decisions; they do not enter the cancellation proof.

The Smagorinsky model approximates the deviatoric subgrid-scale stress using an eddy viscosity determined by the resolved strain-rate tensor:Here ${\bar{S}}_{ij}=({\partial }_j{\bar{u}}_i+{\partial }_i{\bar{u}}_j)/2$, $\Delta$ is the filter length, $C_s$ is dimensionless, and repeated indices use the Einstein summation convention. The Kronecker delta selects the isotropic part. The construction estimates a missing effect from resolved deformation rather than calculating each unresolved eddy.^[ref]

For simple shear flow $\bar{u}=(\gamma y,0,0)$, the only nonzero strain components are ${\bar{S}}_{12}={\bar{S}}_{21}=\gamma /2$, giving $|\bar{S}|=|\gamma |$. Taking $C_s=0.1$, $\Delta =0.02\mathrm{m}$ and $\gamma =10{\mathrm{s}}^{-1}$ yieldsThese are illustrative parameter choices, not a universal fitted constant. The dimensions follow from ${\Delta }^2|\bar{S}|$.

For an incompressible flow, the resolved-to-unresolved kinetic energy transfer isA nonnegative eddy viscosity thus removes resolved kinetic energy and cannot represent local turbulent backscatter. Coefficients and near-wall treatment require physical and numerical judgment.

Joseph Smagorinsky's 1963 atmospheric-circulation experiment used a lateral exchange coefficient based on horizontal deformation and a length scale. The modern three-dimensional expression above extends that strain-dependent modeling idea; it should not be mistaken for the complete original atmospheric model.^[ref]

The subgrid-scale stress records velocity-product information missing from a filtered velocity field. For constant mass density and a specified spatial averaging filter, use the conventionIt enters the filtered momentum equation through $-{\partial }_j{\tau }_{ij}$. This ${\tau }_{ij}$ has units of squared speed; multiplication by mass density gives units of mechanical stress. Some sources use the opposite stress sign, so the momentum equation fixes the convention.^[ref]

For a normalized nonnegative kernel, define $v_i(r)=u_i(x-r)-{\bar{u}}_i(x)$. Expanding the product givesIt is therefore a weighted covariance at the chosen point. For every real vector $b$, $b_i{\tau }_{ij}{b}_j=\int G_{\Delta }(b_i{v}_i{)}^{2}dr\ge 0$, so the stress matrix is positive semidefinite. This argument depends on a nonnegative kernel; it cannot be applied unchanged to filters with negative weights.

For example, average equally weighted velocities $(a,0,0)$ and $(-a,0,0)$. The mean velocity vanishes but $\tau =\mathrm{diag}(a^2,0,0)$. With $a=2\mathrm{m}{\mathrm{s}}^{-1}$, ${\tau }_{11}=4{\mathrm{m}}^2{\mathrm{s}}^{-2}$. A zero mean has hidden nonzero motion.

The definition is exact. A subgrid-scale model instead approximates this tensor using available resolved data. In an incompressible flow, the isotropic part ${\tau }_{kk}{\delta }_{ij}/3$ can be absorbed into a modified pressure; the remaining deviatoric stress controls the modeled directional transfer.

A shell model of turbulence replaces a spatially resolved fluid by a sequence of amplitudes associated with progressively smaller length scales. A typical choice assigns a complex velocity amplitude $u_n$ to the wave-number shell $k_n=k_0{\lambda }^n$, with $\lambda >1$. Neighboring shells exchange energy through nonlinear couplings, while forcing supplies energy and viscosity damps high wave numbers. The model retains a scale cascade at far lower cost than direct numerical simulation, while discarding most spatial geometry.^[ref]

The basic energy bookkeeping is transparent. WriteIf the nonlinear coupling and end conditions satisfy ${\sum }_n\mathrm{Re}(u_n^{\ast }{N}_n)=0$, differentiating givesThe nonlinearity transfers energy between shells without creating its total. Viscous damping becomes faster at larger $k_n$: an isolated unforced linear mode decays as $e^{-\nu k_n^{2}t}$.

For an illustrative constant-flux estimate, a transfer time ${\tau }_n\sim (k_n|u_n|{)}^{-1}$ gives $\epsilon \sim |u_n{|}^2/{\tau }_n\sim k_n|u_n{|}^3$, hence $|u_n|\sim (\epsilon /k_n{)}^{1/3}$. Deviations of higher moments from a single scaling rule can then diagnose intermittency. The estimate is dimensional reasoning, not an exact solution. The published improved coupling was designed to reduce spurious shell correlations while preserving useful conservation and scaling properties.^[ref]

A structure function measures moments of a field's differences across a separation. For a velocity field $u$, define the longitudinal incrementThe average may be over an ensemble or over a suitable homogeneous sample. Stating the component, separation direction and absolute-value convention matters: signed odd moments contain information that absolute moments erase.^[ref]

An independent smooth-field example is $u(x)=U\sin (kx)$ in one dimension, averaged over a whole period. Direct integration givesThe quadratic small-separation behavior records differentiability. A constant offset added to the velocity disappears from every increment.

The signed third moment has a special physical role. In the inertial range of high-Reynolds-number, locally isotropic, incompressible turbulence under the usual stationary energy-balance assumptions,with $\epsilon$ the mean dissipation rate per unit mass. This relation concerns a signed cubic increment, not $S_3$ as defined with an absolute value above. Its sign records a directed transfer that a second-order spectrum cannot determine. At finite viscosity the energy-balance relation has an additional derivative term; anisotropic, transient or compressible data need their own justification before this inertial-range formula is applied.^[ref]

Intermittency is the uneven concentration of turbulent fluctuations or dissipation into relatively intense events separated by quieter regions or intervals. It often appears statistically as increasingly non-Gaussian velocity or magnetic-field increments at smaller separations. A turbulent signal can therefore have a moderate average variance while rare events dominate its higher moments.^[ref]

One useful diagnostic is the increment flatnessusing the same structure function component and convention in both moments. A centered Gaussian has $F=3$. Increasing $F$ towards small scales indicates increasingly strong relative tails, although measurement noise and nonstationary mixtures also need checking.

A toy distribution isolates the rare-event effect. Let $X$ vanish with probability $1-p$ and take values $+\sigma /\sqrt{p}$ and $-\sigma /\sqrt{p}$ with probability $p/2$ each. ThenAt $p=0.05$, the variance is still ${\sigma }^2$ but the flatness is $20$. This is an illustrative marginal distribution, not a construction of a complete turbulent field.

Spacecraft measurements provide a concrete application. Analysis of Helios 2 magnetic and velocity records compared increments at different lags and heliocentric distances, finding scale-dependent flatness in both fast and slow solar wind. Mapping time lag to spatial separation requires an advection assumption. Flatness characterizes the fluctuations; by itself it does not establish the sign or magnitude of an energy cascade.^[ref]

Reynolds averaging separates a resolved mean from zero-mean fluctuations. The idealized averaging operation is linear and idempotent, commutes with the derivatives under consideration and treats resolved mean factors as constant inside its averaging scale. Products then satisfy $\overline{fg}=\overline{f}\overline{g}+\overline{f^{\prime }{g}^{\prime }}$. Actual finite spatial filters need scale separation or corrections when these rules are only approximate.

A fixed observation point alternates between turbulent and non-turbulent fluid because a fluctuating interface passes it. This is distinct from internal intermittency, which concerns variations within fluid that remains turbulent.

Wave turbulence describes the statistical transfer of energy between interacting waves. The wave period, nonlinear interaction time, phase correlations and resonances determine the transfer rate. A weak wave cascade treats the change in one interaction as small; critical balance instead makes the nonlinear interaction rate comparable to the linear wave frequency.

If a wave packet experiences a fractional nonlinear change $\chi ={\tau }_w/{\tau }_{\mathrm{nl}}\ll 1$ per interaction, and successive interaction phases are decorrelated, $N$ changes add as a random walk of size $\sqrt{N}\chi$. Order-one transfer needs $N\sim {\chi }^{-2}$ interactions, giving the displayed cascade time. Phase coherence or resonant constraints can alter this estimate; weakness alone does not justify random addition.

Magnetohydrodynamic turbulence is irregular nonlinear motion of an electrically conducting fluid coupled to a magnetic field. Alfvén waves and magnetosonic waves give propagation times in addition to the nonlinear mixing time. A strong guide field distinguishes directions parallel and perpendicular to it, so the resulting cascades need not be isotropic.

In a local isotropic hydrodynamic inertial-range model with only $ϵ$ and $k$, dimensional analysis fixes the Kolmogorov 1941 theory spectrum. A mean magnetic field supplies the independent Alfvén speed, permitting the arbitrary dimensionless function in the displayed formula even before anisotropy adds parallel scales. Cascade-time assumptions such as weak decorrelated interactions or critical balance are therefore necessary to select an MHD spectrum.

An electron-magnetohydrodynamic cascade transfers magnetic fluctuations through the nonlinear electron magnetohydrodynamics induction equation. Writing $b_l=\delta B_l/B_0$, its perpendicular nonlinear interaction time is ${\tau }_{\mathrm{nl}}\sim l_{\perp }^2/(v_A{d}_i{b}_l)$. The anisotropic whistler wave time is ${\tau }_w\sim l_{\parallel }{l}_{\perp }/(v_A{d}_i)$. Scaling predictions depend on locality, phase decorrelation and the assumed balance of these times.

Critical balance sets ${\tau }_{\mathrm{nl}}\sim {\tau }_w$, hence $l_{\parallel }\sim l_{\perp }/b_l$. A local strong cascade with constant energy flux has $ϵ\sim v_A^3{d}_i{b}_l^3/l_{\perp }^2$, giving the displayed exponents and perpendicular magnetic energy spectrum $E_B(k_{\perp })\propto k_{\perp }^{-7/3}$. The anisotropy becomes stronger towards small perpendicular scales.

For $b_l{l}_{\parallel }/l_{\perp }\ll 1$, the weak wave cascade estimate gives ${\tau }_{\mathrm{cas}}\sim l_{\perp }^3/(v_A{d}_i{b}_l^2{l}_{\parallel })$. Constant energy flux $ϵ\sim v_A^2{b}_l^2/{\tau }_{\mathrm{cas}}$ yields the displayed amplitude. An additionally isotropic local cascade has $b_l\propto l^{1/2}$ and shell-integrated magnetic energy spectrum $E_B(k)\propto k^{-2}$. These statements are phenomenological consequences of the chosen locality and decorrelation assumptions.

Critical balance is a scale-dependent ordering in which the wave-propagation time and nonlinear interaction time are comparable. For strong Alfvénic turbulence, $k_{\parallel }{v}_A\sim k_{\perp }\delta u_{\perp }$. For electron magnetohydrodynamics, the linear rate is $v_A{d}_i{k}_{\parallel }{k}_{\perp }$ and the nonlinear rate is $v_A{d}_i{k}_{\perp }^2\delta B/B_0$, so critical balance instead gives $\delta B/B_0\sim k_{\parallel }/k_{\perp }$. This assumption is an estimate of decorrelation dynamics, not an exact equality at every point.

Balance the inertial wave rate against horizontal nonlinear advection. The anisotropy $ϵ=|k_{\parallel }|/k_{\perp }$ is then of the same order as the Rossby number $k_{\perp }U/(2\Omega )$. Horizontal geostrophic flow gives kinematic pressure amplitude $\delta p\sim 2\Omega U/k_{\perp }\sim U^2/ϵ$. The first ageostrophic flow is $O(ϵU)$ and supplies the small horizontal divergence balancing $-{\partial }_z{u}_{\parallel }$. Rapid inertial oscillations are excluded by this slow-time ordering.

Alfvénic turbulence is the turbulent interaction of fluctuations polarized primarily as Alfvén waves. In Elsässer variables ${\mathbf{z}}^{\pm }={\mathbf{u}}_{\perp }\pm \delta {\mathbf{B}}_{\perp }/\sqrt{4\pi {\rho }_0}$, each wave population is nonlinearly distorted by the oppositely propagating population. Equal populations give a balanced cascade; unequal populations give an imbalanced cascade. The absence of one population removes the leading nonlinear interaction in ideal reduced magnetohydrodynamics.

For balanced anisotropic Alfvénic turbulence, suppose ${\tau }_A=l_{\parallel }/v_A\ll {\tau }_{\mathrm{nl}}=l_{\perp }/\delta z$ and resonant interactions preserve the parallel scale during the perpendicular cascade. Random encounter accumulation gives ${\tau }_{\mathrm{cas}}=v_A{l}_{\perp }^2/(\delta z^2{l}_{\parallel })$. Constant energy flux then gives $\delta z\propto l_{\perp }^{1/2}$ and the displayed perpendicular turbulent energy spectrum. Unlike isotropic phenomenology, the fixed parallel scale is independent of the decreasing perpendicular scale.

In a weak Alfvénic cascade at fixed $l_{\parallel }$, $\chi ={\tau }_A/{\tau }_{\mathrm{nl}}\propto l_{\perp }^{-1/2}$ grows as the perpendicular scale shrinks. If its outer value is ${\chi }_0\ll 1$ at $L_{\perp }$, it reaches order one at the displayed scale. Beyond it the weak-interaction ordering fails and a critical balance model may become appropriate.

The Iroshnikov-Kraichnan spectrum phenomenology assumes balanced, isotropic local Alfvénic turbulence with $\delta z_l\ll v_A$, decorrelated counterpropagating encounters, ${\tau }_A=l/v_A$ and ${\tau }_{\mathrm{nl}}=l/\delta z_l$. A weak wave cascade takes ${\tau }_{\mathrm{cas}}\sim {\tau }_{\mathrm{nl}}^2/{\tau }_A=v_{A}l/\delta z_l^2$. Constant energy flux gives $\delta z_l^4\sim ϵv_{A}l$ and the displayed shell-integrated turbulent energy spectrum. Isotropy is an additional assumption, not a consequence of a strong guide field.

Goldreich–Sridhar turbulence combines critical balance, balanced counterpropagating Alfvén waves, a local perpendicular cascade, and constant energy flux per unit mass $\mathcal{E}$. With ${\tau }_{\mathrm{nl}}\sim {\ell }_{\perp }/\delta u_{\ell }$ and $\mathcal{E}\sim \delta u_{\ell }^2/{\tau }_{\mathrm{nl}}$, one obtains $\delta u_{\ell }\sim (\mathcal{E}{\ell }_{\perp }{)}^{1/3}$. Defining the one-dimensional perpendicular turbulent energy spectrum by $\delta u_{\ell }^2\sim k_{\perp }E(k_{\perp })$ gives $E(k_{\perp })\sim {\mathcal{E}}^{2/3}{k}_{\perp }^{-5/3}$. Critical balance also gives $k_{\parallel }\sim {\mathcal{E}}^{1/3}{k}_{\perp }^{2/3}/v_A$. This scaling argument neglects scale-dependent alignment and intermittency corrections, which are additional physical effects rather than consequences of the stated assumptions.

A turbulence closure approximates the unclosed higher-order moments in evolution equations for lower-order turbulent statistics. For example, an equation for the velocity correlation tensor involves triple correlations, whose evolution involves fourth-order correlations and pressure. A closure must be checked against conservation laws, realizability and boundary conditions; matching an energy spectrum alone does not establish all these properties.

This turbulence closure assumes the velocity increment skewness is the same throughout the equilibrium range, with its value fixed by the Kolmogorov two-thirds law and Kolmogorov four-fifths law. It yields a simple crossover between differentiable and inertial-scale increments, but suppresses the scale dependence of skewness and has no independent spectral-transfer dynamics.

Set $S_2=\sqrt{15}{\beta }^{3/2}{v}_{\eta }^{2}h$ and $r=(15\beta {)}^{3/4}\eta x$. The Kolmogorov equation for structure functions and the constant-skewness turbulence closure give the displayed ordinary differential equation, with $h(0)=0$. Its small-separation balance is $h\sim x^2$; its large-separation balance is $h\sim x^{2/3}$. The first matches differentiable velocity increments, and the second recovers the assumed Kolmogorov two-thirds law.

EDQNM closure models the evolution of the turbulent energy spectrum through spectral triad interactions. Fourth-order correlations are approximated using products of second-order correlations, an eddy damping limits the resulting triple correlations, and their memory integrals are replaced by expressions using current spectra. The modeled damping times are additional physical input. This allows evolving spectra and scale-dependent transfer, but does not automatically recover coherent structures or internal intermittency. Bertoglio, Squires and Ferziger's simulation comparison explains the damping assumptions and tests their limitations.

A two-equation closure transporting turbulent kinetic energy $k$ and its turbulent kinetic energy dissipation rate $ϵ$. Its eddy viscosity is $C_{\mu }{k}^2/ϵ$. In a high-Reynolds-number wall-normal reduction, turbulent diffusion has coefficients ${\nu }_T/{\sigma }_k$ and ${\nu }_T/{\sigma }_{ϵ}$; the energy equation has shear production $P$ and destruction $ϵ$, while the dissipation equation has modeled production $C_{ϵ1}Pϵ/k$ and destruction $C_{ϵ2}{ϵ}^2/k$. The dissipation production/destruction terms are closure terms, not a second conservation of energy.

A stationary turbulence edge with no production has an advection–turbulent-diffusion leading balance. For $k\propto x^p$, $ϵ\propto x^q$ and ${\nu }_T=C_{\mu }{k}^2/ϵ$, this balance gives $2p-q=1$ and $q/p={\sigma }_{ϵ}/{\sigma }_k=\sigma$. The integral length $k^{3/2}/ϵ$ and turnover frequency $ϵ/k$ both tend to zero when $1<\sigma <3/2$. The front viscosity is linear, with ${\nu }_T^{\prime }(0^{+})=(2{\sigma }_k-{\sigma }_{ϵ})U_0$. This is formal high-Reynolds-number consistency, not a uniform justification for neglecting molecular viscosity at the exact edge.

In the constant-stress overlap layer, the law of the wall gives $U_y=u_{\ast }/(\kappa y)$ and the Reynolds stress gives $-\overline{u^{\prime }{v}^{\prime }}\simeq u_{\ast }^2$. Thus ${\nu }_T=u_{\ast }\kappa y$ and local balance gives $P\simeq ϵ=u_{\ast }^3/(\kappa y)$. Combining with the K-epsilon model viscosity formula gives the displayed constant $k$. The energy diffusion term then vanishes. The dissipation diffusion term does not: it is $u_{\ast }^4/({\sigma }_{ϵ}{y}^2)$. Balancing it against the dissipation-equation source terms yields the displayed Von Kármán constant relation. Neglecting dissipation diffusion would incorrectly require equal production and destruction constants.

Turbulent mixing redistributes a transported scalar, such as temperature or particle concentration, through irregular fluid motions and eventual molecular diffusion. A top-hat plume model assumes this mixing keeps the scalar approximately uniform within a selected cross section; that uniformity is an additional closure, not a consequence of thin geometry alone.

Mixing across surfaces of constant density, or isopycnals. In the deep ocean, breaking internal waves can supply the mechanical energy needed to mix against stable stratification. The divergence of the turbulent buoyancy flux determines local water-mass transformation.

A planar turbulent mixing layer develops between two parallel streams of different mean velocities. Their shear flow generates turbulence that transports streamwise linear momentum across the interface. A thin-layer eddy viscosity model with ${\nu }_T=C\Delta U\delta (x)$ has thickness proportional to downstream distance. For $\Delta U/U_c\ll 1$, its leading mean velocity transition is an error function.

For a planar turbulent mixing layer, let $U_c=(U_a+U_b)/2$, $\Delta U=U_a-U_b$, ${\nu }_T=C\Delta U\delta$, $\delta =S\Delta Ux/U_c$, and $\xi =\sqrt{S/C}y/\delta$. The stream function $\psi =U_c[y+\delta \sqrt{C/S}F(\xi )]$ enforces mean incompressibility. Substitution into $U{U}_x+V{U}_y={\nu }_T{U}_{yy}$ gives the displayed ordinary differential equation, with $F^{\prime }(\pm \infty )=\pm \Delta U/(2U_c)$. When $\Delta U/U_c$ is small, its linear approximation is $F^{\prime \prime \prime }+\xi F^{\prime \prime }=0$, so $F^{\prime \prime }$ is a Gaussian function and $F^{\prime }$ an error function.