In aeroacoustics, spectral broadening refers to the scattering of tonal sound fields by turbulent shear layers, whereby the interaction of the sound with turbulent flow results in power lost from the tone and distributed into a broadband field around the tone frequency. Fan and turbine tone spectral broadening is known colloquially as "haystacking". Recently a new analytical model has been derived to predict weak spectral broadening of a tone radiated through a circular jet. A key part of the modeling is the choice of the two-point turbulent velocity cross-correlation function which is used to provide a statistical description of the turbulence in the shear layer. A new cross-correlation function for an axisymmetric turbulent shear layer formed by a circular jet, based on the theory for homogeneous axisymmetric turbulence, has been developed. Validation results of weak-scattering calculated using this correlation function show better agreement with measurements when compared to the results calculated using a correlation function based on the theory for homogeneous isotropic turbulence.Doppler factor, incident field [Hz] D = Doppler factor, scattered field [Hz] e (1) , e (2) = unit vectors defining orthogonal coordinate system f = frequency, incident field [Hz] F = frequency, scattered field [Hz] . AIAA Senior member.k = axial wavenumber, incident field [radm −1 ] K = axial wavenumber, scattered field [radm −1 ] L = integral lengthscale [m] m = azimuthal order, incident field M = azimuthal order, scattered field p = pressure [Pa] r c = jet radius [m] r = separation vector [m] R = specific gas constant [Jkg −1 K −1 ] R i j = turbulent velocity two-point cross-correlation R i j = turbulent velocity two-point cross-correlation (without temporal decay) s = azimuthal separation r µ φ [m] t = time [s] T = temperature [K] T = integral timescale [s] u = velocity (u, v, w) [ms −1 ] U c = turbulence convection velocity [ms −1 ] U J = core jet flow velocity [ms −1 ] U (r) = mean velocity profile [ms −1 ] (x,r, φ) = polar coordinate system (x, y, z) = Cartesian coordinate system Greek variables β = non-dimensional constant used to define temporal decay function γ = adiabatic constant C p /C V γ m = radial wavenumber, incident field [radm −1 ] Γ M = radial wavenumber, scattered field [radm −1 ] δ = shear layer thickness [m] θ = polar angle, incident field [rad] Θ = polar angle, scattered field [rad] 2 λ = unit vector defining axis of symmetry µ = spatial separation distance [m] ρ = density [kgm −3 ] τ = temporal separation time [s] Φ i j = turbulent velocity cross-spectrum ω = angular frequency, incident field [rads −1 ] Ω = angular frequency, scattered field [rads −1 ] Subscripts d = denotes directivity of the incident field D = denotes directivity of the scattered field i, j = indices denoting x, r or φ m = denotes azimuthal order of the incident field M = denotes azimuthal order of the scattered field s = denotes sound (acoustic) quantity t = denotes turbulent quantity ∞ = denotes value at infinity Superscripts = denotes pe...