Operators

AbstractDerivedStructureFunctionType

A quantity derived from binned structure functions, with no value on one pair.

AbstractPairwiseStructureFunctionType

An operator with a value on one pair, sf(δu, r̂), that the pair loops, the gridded sweeps and the transforms accumulate.

AbstractStructureFunctionType

The operator of a structure function: which statistic of a pair's increment is accumulated.

ProjectedStructureFunctionType{3, 0}: the longitudinal third-order structure function $⟨δu_L³⟩$.

ProjectedStructureFunctionType{2, 1}: $⟨δu_L² δu_T⟩$ with the signed transverse component of the operator's basis.

DivergentSecondOrderStructureFunctionType()

Helmholtz-derived 2D divergent second-order component. A derived quantity, computed from binned L2SF/T2SF with helmholtz_decompose_2d.

Alias of FullVectorStructureFunctionType.

FullVectorStructureFunctionType{NF}

Generic full-vector norm-power operator, $||δu||^NF$. S3SF's conventional definition is ThirdOrderStructureFunctionType.

HelmholtzDecomposition2DType()

Derived quantity describing the 2D isotropic Helmholtz decomposition into rotational and divergent second-order components.

Shorthand for LongitudinalTransverseComponentThirdOrderStructureFunctionType.

Shorthand for OffDiagonalInconsistentThirdOrderStructureFunctionType, $⟨δu_L ‖δu_T‖²⟩$.

Shorthand for LongitudinalSecondOrderStructureFunctionType, $⟨δu_L²⟩$.

Shorthand for DiagonalInconsistentThirdOrderStructureFunctionType, $⟨δu_L² δu_T⟩$.

Shorthand for DiagonalConsistentThirdOrderStructureFunctionType, $⟨δu_L³⟩$.

ProjectedStructureFunctionType{2, 0}: the longitudinal second-order structure function $⟨δu_L²⟩$.

LongitudinalTransverseComponentThirdOrderStructureFunctionType()

Per-component variant of L1T2SF, $δu_L * ||δu_t||² / (D - 1)$.

Shorthand for MixedStructureFunctionType.

MixedStructureFunctionType{NL, NT, P}(vector_field = 1, scalar_field = 1)

$⟨δu_L^{NL} ‖δu_T‖^{NT} (δθ)^P⟩$ — a velocity–scalar mixed moment.

{1, 0, 2} is Yaglom's law, $⟨δu_L (δθ)²⟩ = −(4/3) ε_θ r$; {1, 0, 1} is the flux of the tracer itself. The velocity part is read from a transported vector field and the scalar part from a differenced scalar field, so the two never mix frames.

MomentTensorOperator{P}()

The transform engine's request for the whole rank-P increment moment tensor M[i₁, …, i_P] = Σ_pairs Π_k δu[i_k] of a field's vector field, in place of one scalar contraction of it. It has no value on a single pair. In a fixed Cartesian frame an odd rank changes sign when a pair is read from its other end, exactly as an odd scalar increment does, so it takes the same canonical pair reading; in a pair's own geodesic frame the components are read the same from either end and no reading enters.

ProjectedStructureFunctionType{0, 3}: $⟨δu_T³⟩$ with the signed transverse component of the operator's basis.

ProjectedStructureFunctionType{1, 2}: $⟨δu_L ‖δu_T‖²⟩$.

Alias of ProjectedStructureFunctionType.

ProjectedStructureFunctionType{NL, NT}(basis = CanonicalTransverseBasis())

Operator δu_L^NL · δu_T^NT: NL powers of the longitudinal increment δu·r̂ and NT of the transverse one. NT = 2 is the invariant transverse energy ‖δu‖² − δu_L²; every other NT is the signed component along the first vector of transverse_basis(basis, r̂), so the sign of an odd NT is the convention's. SFH.CanonicalTransverseBasis is n̂, the turn about ẑ. The basis travels with the operator into every result.

The two readings differ only for D ≥ 3, where the transverse plane has more than one direction; in 2-D ‖δu_T‖² = (δu·n̂)². They cannot be unified: ‖δu_T‖^NT is a polynomial in δu only for even NT, since ‖δu_T‖² = ‖δu‖² − δu_L² and an odd power needs the square root, so odd NT admits the component reading alone. NT = 2 is the energy because that is the quantity the second-order relations are written in — the Helmholtz split and the D_LL/D_TT isotropy relations all take it. TransverseComponentSecondOrderStructureFunctionType is the per-component form, ‖δu_T‖²/(D−1).

(sf::ProjectedStructureFunctionType{NL,NT})(δu, r̂)

Compute the structure function kernel for longitudinal/transverse components.

• NL : power of longitudinal component δu_l
• NT : power of transverse component ||δu_t||

RotationalSecondOrderStructureFunctionType()

Helmholtz-derived 2D rotational second-order component. A derived quantity, computed from binned L2SF/T2SF with helmholtz_decompose_2d.

Shorthand for SecondOrderStructureFunctionType, $⟨‖δu‖²⟩$.

Shorthand for ThirdOrderStructureFunctionType, $⟨δu_L ‖δu‖²⟩$.

ScalarDotStructureFunctionType(a, b)

$⟨δθ^{(a)} δθ^{(b)}⟩$ — a second-order cross-field scalar moment.

(1, 1) is the scalar structure function. (a, b) with a ≠ b is what an advective structure function is: ⟨δω δ𝓐_ω⟩ is this with ω and its advection as the two fields.

Shorthand for ScalarStructureFunctionType.

ScalarStructureFunctionType{P}(field = 1)

$⟨(δθ)^P⟩$ on scalar field — the scalar structure function. P = 2 is the Obukhov–Corrsin quantity; odd P measures the skewness of the tracer increment.

SecondOrderStructureFunctionType()

Full-vector second-order structure function, $S2SF = ||δu||²$.

Shorthand for TransverseComponentSecondOrderStructureFunctionType.

Shorthand for TransverseSecondOrderStructureFunctionType, $⟨‖δu_T‖²⟩$.

Shorthand for OffDiagonalConsistentThirdOrderStructureFunctionType, $⟨δu_T³⟩$.

ThirdOrderStructureFunctionType()

Third-order scalar flux structure function, $S3SF = δu_L * ||δu||² = L3SF + L1T2SF$. It is intentionally not $||δu||³$.

TransverseComponentSecondOrderStructureFunctionType()

Per-component transverse second-order structure function, $||δu_t||² / (D - 1)$. This is distinct from T2SF, which stores the total transverse energy.

ProjectedStructureFunctionType{0, 2}: the transverse second-order structure function $⟨‖δu_T‖²⟩ = ⟨‖δu‖² − δu_L²⟩$.

VectorDotStructureFunctionType(a, b)

$⟨δu^{(a)} · δu^{(b)}⟩$ — a second-order cross-field vector moment.

(1, 1) is S2SF: the existing second-order operator is this one's diagonal, not a separate thing. (a, b) with a ≠ b is ⟨δu · δ𝓐_u⟩, the advective structure function, which holds without isotropy — the reason it is worth having beside the third-order laws.

The instance DiagonalConsistentThirdOrderStructureFunctionType().

The instance DiagonalInconsistentThirdOrderStructureFunctionType().

The instance DivergentSecondOrderStructureFunctionType().

The instance HelmholtzDecomposition2DType().

The instance LongitudinalTransverseComponentThirdOrderStructureFunctionType().

The instance L1T2SFType().

The instance L2SFType().

The instance L2T1SFType().

The instance L3SFType().

The instance LongitudinalSecondOrderStructureFunctionType().

The instance OffDiagonalConsistentThirdOrderStructureFunctionType().

The instance OffDiagonalInconsistentThirdOrderStructureFunctionType().

The instance RotationalSecondOrderStructureFunctionType().

The instance S2SFType().

The instance S3SFType().

The instance SecondOrderStructureFunctionType().

The instance TransverseComponentSecondOrderStructureFunctionType().

The instance T2SFType().

The instance T3SFType().

The instance ThirdOrderStructureFunctionType().

The instance TransverseSecondOrderStructureFunctionType().

get_structure_function_type(order::Int, mode::Symbol)

Map an integer order and a mode symbol (e.g., :longitudinal, :transverse, :scalar/total) to a specific operator instance.