Forcing

Forcing

Defines methods for injecting energy onto a system of vortices.

NormalFluidForcing <: AbstractForcing
NormalFluidForcing(vn::Function; α, α′ = 0)

Forcing due to mutual friction with a normal fluid.

The normal fluid is represented by a function vn which should take a position x⃗::SVector{N, T} and return a velocity v⃗::SVector{N, T} (N is the number of dimensions, usually N = 3).

In particular, the function could be a synthetic velocity field from the SyntheticFields module (see below for examples).

This type of forcing defines an external velocity $\bm{v}_{\text{f}}$ affecting vortex motion, so that the actual vortex velocity $\bm{v}_{\text{L}}$ is

\[\frac{\mathrm{d}\bm{s}}{\mathrm{d}t} = \bm{v}_{\text{L}} = \bm{v}_{\text{s}} + \bm{v}_{\text{f}}.\]

Here $\bm{v}_{\text{s}}$ is the self-induced velocity obtained by applying Biot–Savart's law.

The forcing velocity is of the form:

\[\bm{v}_{\text{f}} = α \bm{s}' × \bm{v}_{\text{ns}} - α' \bm{s}' × \left( \bm{s}' × \bm{v}_{\text{ns}} \right)\]

where $\bm{s}'$ is the local unit tangent vector, $\bm{v}_{\text{ns}} = \bm{v}_{\text{n}} - \bm{v}_{\text{s}}$ is the local slip velocity, and $α$ and $α'$ are non-dimensional coefficients representing the intensity of Magnus and drag forces.

Example

Define a mutual friction forcing based on a large-scale normal fluid velocity field (see SyntheticFields.FourierBandVectorField):

julia> using VortexPasta.Forcing: NormalFluidForcing

julia> using VortexPasta.SyntheticFields: SyntheticFields, FourierBandVectorField

julia> using Random: Xoshiro

julia> rng = Xoshiro(42);  # initialise random number generator (optional, but recommended)

julia> Ls = (2π, 2π, 2π);  # domain dimensions

julia> vn_rms = 1.0;  # typical magnitude (rms value) of normal fluid velocity components

julia> vn = FourierBandVectorField(undef, Ls; kmin = 0.1, kmax = 1.5)  # create field with non-zero Fourier wavevectors kmin ≤ |k⃗| ≤ kmax
FourierBandVectorField{Float64, 3} with 9 independent Fourier coefficients in |k⃗| ∈ [1.0, 1.4142]

julia> SyntheticFields.init_coefficients!(rng, vn, vn_rms);  # randomly set non-zero Fourier coefficients of the velocity field

julia> forcing = NormalFluidForcing(vn; α = 0.8, α′ = 0)
NormalFluidForcing{Float64} with:
 ├─ Normal velocity field: FourierBandVectorField{Float64, 3} with 9 independent Fourier coefficients in |k⃗| ∈ [1.0, 1.4142]
 └─ Friction coefficients: α = 0.8 and α′ = 0.0

Forcing.apply!(forcing::AbstractForcing, vs::AbstractVector{<:Vec3}, f::AbstractFilament; [scheduler])

Apply forcing to a single filament f with self-induced velocities vs.

At output, the vs vector is overwritten with the actual vortex line velocities.

The optional scheduler keyword can be used to parallelise computations using one of the schedulers defined in OhMyThreads.jl.

Forcing.apply!(forcing::NormalFluidForcing, vs, vn, tangents; [scheduler])

This variant can be used in the case of a NormalFluidForcing if one already has precomputed values of the normal fluid velocity and local unit tangents at filament points.

FourierBandForcing <: AbstractForcing
FourierBandForcing(vn::FourierBandVectorField; α, α′ = 0, filtered_vorticity = false)

Forcing due to mutual friction of a normal fluid with a Fourier-filtered superfluid velocity.

This forcing is similar to NormalFluidForcing, but tries to only affect scales within a given band [kmin, kmax] in Fourier space. This is achieved by a normal fluid velocity field represented by a FourierBandVectorField, and by a modified Schwarz's equation in which only a coarse-grained superfluid flow is taken into account in the estimation of the mutual friction term.

Concretely, the vortex line velocity according to this forcing type is:

\[\frac{\mathrm{d}\bm{s}}{\mathrm{d}t} = \bm{v}_{\text{L}} = \bm{v}_{\text{s}} + \bm{v}_{\text{f}}\]

The forcing velocity is of the form:

\[\bm{v}_{\text{f}} = α \bm{s}' × \bm{v}_{\text{ns}}^{>} - α' \bm{s}' × \left( \bm{s}' × \bm{v}_{\text{ns}}^{>} \right)\]

where $\bm{v}_{\text{ns}}^{>} = \bm{v}_{\text{n}} - \bm{v}_{\text{s}}^{>}$ is a filtered slip velocity. In practice, the filtered velocity is active within the same wavenumber range [kmin, kmax] where vn is defined. See NormalFluidForcing for other definitions.

Using a filtered vorticity field

To further ensure that this forcing only affects the chosen range of scales, one can pass filtered_vorticity = true, which will replace the local unit tangent $\bm{s}'$ with a normalised coarse-grained vorticity. This corresponds to setting $\bm{s}' = \bm{ω}^{>} / |\bm{ω}^{>}|$ where $\bm{ω}^{>}$ is the Fourier-filtered vorticity field.

AbstractForcing

Abstract type representing a forcing method.