Forcing

Forcing

Defines methods for injecting and dissipating the energy generated by a system of vortices.

NoForcing() <: AbstractForcing

Represents the absence of external forcing.

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

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.

FourierBandForcingBS <: AbstractForcing
FourierBandForcingBS(; kmin, kmax, α, ε_target)

Forcing based on Biot–Savart energetics within a given range of wavenumbers.

This type of forcing does not rely on an imposed normal fluid velocity. Instead, it starts from the functional derivative of the energy at a given wavevector $\bm{k}$ with respect to the vortex positions (according to the Biot–Savart law), and applies a velocity that ensures energy to increase (if $α > 0$ or $ε_{\text{target}} > 0$) at those wavevectors.

One should pass either α or ε_target but never both. They should be positive for energy injection (negative values lead to energy dissipation):

• α (\alpha) is a non-dimensional coefficient which directly sets the amplitude of the forcing velocity;

• ε_target (\varepsilon_target) has the units of an energy injection rate. In this case, the amplitude $α$ will be adjusted over time in order to keep a roughly constant energy injection rate (which in general will not be equal to ε_target, see remarks below).

Extended help

Forcing definition

This forcing attempts to increase the kinetic energy at selected wavenumbers. Its definition starts from the expression for the kinetic energy at wavenumber $\bm{k}$:

\[E(\bm{k}) = \frac{1}{2} |\bm{v}(\bm{k})|^2 = \frac{1}{2k^2} |\bm{ω}(\bm{k})|^2\]

The idea is to translate the vortex positions by $\bm{s}(ξ) → \bm{s}(ξ) + δ\bm{s}(ξ)$ so that the energy $E(\bm{k})$ increases. To determine such a displacement, one can look at the functional derivative of $E(\bm{k})$ with respect to the positions $\bm{s}$:

\[\begin{align*} \frac{δE(\bm{k})}{δ\bm{s}} &= \frac{1}{2k^2} \bm{ω}(\bm{k}) ⋅ \frac{δ\bm{ω}^*(\bm{k})}{δ\bm{s}} + \text{c.c.}, \\ &= \frac{1}{2} \bm{ψ}(\bm{k}) ⋅ \frac{δ\bm{ω}^*(\bm{k})}{δ\bm{s}} + \text{c.c.}, \end{align*}\]

where $()^*$ denotes a complex conjugate and "c.c." means the complex conjugate of the first term. Here we have used $\bm{ω}(\bm{k}) = k^2 \bm{ψ}(\bm{k})$, which corresponds in physical space to $\bm{ω} = -∇² \bm{ψ}$.

The functional derivative of vorticity in Fourier space is:

\[\begin{align*} \frac{δ\bm{ω}^*(\bm{k})}{δ\bm{s}} &= \frac{Γ}{V} \frac{δ}{δ\bm{s}} ∮ e^{+i \bm{k} ⋅ \bm{s}(ξ)} \bm{s}(ξ)' \, \text{d}ξ \\ &= \frac{Γ}{V} \frac{δ}{δ\bm{s}} ∮ \bm{ℒ}[\bm{s}(ξ), \bm{s}(ξ)'] \, \text{d}ξ \\ &= \frac{Γ}{V} \left[ \frac{∂\bm{ℒ}}{∂\bm{s}} - \frac{\text{d}}{\text{d}ξ} \frac{∂\bm{ℒ}}{∂\bm{s}'} \right] \\ &= \frac{i Γ}{V} e^{+i \bm{k} ⋅ \bm{s}} \left[ \bm{k} ⊗ \bm{s}' - (\bm{k} ⋅ \bm{s}') I \right] = \frac{i Γ}{V} \bm{B}(\bm{k}, \bm{s}, \bm{s}') \end{align*}\]

where $\bm{B}$ is a $3×3$ complex matrix which has dimensions of an inverse length ($L^{-1}$).

The functional derivative of $E(\bm{k})$ can now be written as:

\[\begin{align*} \frac{δE(\bm{k})}{δ\bm{s}} &= \frac{Γ}{2V} i \bm{B} \bm{ψ}(\bm{k}) + \text{c.c.} \\ &= \frac{Γ}{2V} [\bm{s}' × \bm{v}(\bm{k})] e^{i \bm{k} ⋅ \bm{s}} + \text{c.c.} \\ &= ℜ \left\{ \frac{Γ}{V} [\bm{s}' × \bm{v}(\bm{k})] e^{i \bm{k} ⋅ \bm{s}} \right\} \end{align*}\]

Finally, the idea is to advect the filaments with a velocity which is parallel to this result for each forced $\bm{k}$. One can write this velocity as

\[\bm{v}_{\text{f}}(\bm{k}, \bm{s}) = α \, ℜ \left\{ [\bm{s}' × \bm{v}(\bm{k})] e^{i \bm{k} ⋅ \bm{s}} \right\} = α \, \bm{v}_0(\bm{k}, \bm{s})\]

where $α$ is a non-dimensional parameter setting the forcing amplitude.

Estimating the energy injection rate

One can also try to estimate an energy injection rate at wavevector $\bm{k}$ associated to this velocity:

\[\frac{\text{d}E(\bm{k})}{\text{d}t} = ∮ \frac{δE(\bm{k})}{δ\bm{s}} ⋅ \frac{\text{d}\bm{s}}{\text{d}t} \, \mathrm{d}ξ = α \frac{Γ}{V} ∮ |\bm{v}_0|^2 \, \mathrm{d}ξ\]

In general, this estimate may be quite inaccurate since the forcing can also affect the energy at wavevectors other than $\bm{k}$. But still, this estimate is the one used when $ε_{\text{target}}$ is given, and may allow to obtain a roughly constant energy injection rate (even if it's generally different than the "target" one).

NoDissipation() <: AbstractDissipation

Repersents the absence of dissipation term.

SmallScaleDissipationBS <: AbstractDissipation
SmallScaleDissipationBS(; kdiss, α, ε_target)

Dissipation based on Biot–Savart energetics at large wavenumbers (small scales).

This dissipation method works similarly to FourierBandForcingBS. The main difference is that it acts at a different range of scales (wavenumbers k > kdiss), and that positive values of α or ε_target lead to dissipation and not injection.

As in FourierBandForcingBS, one should pass either α or ε_target but never both. They should be positive for energy dissipation (negative values lead to energy injection at small scales, which may lead to instabilities!):

• α (\alpha) is a non-dimensional coefficient which directly sets the amplitude of the dissipation term;

• ε_target (\varepsilon_target) has the units of an energy dissipation rate. In this case, the amplitude $α$ will be adjusted over time in order to keep a roughly constant energy dissipation rate (which in general will not be equal to ε_target, as discussed in FourierBandForcingBS).

Extended help

Dissipation term definition

The idea is to apply a "dissipation velocity" $\bm{v}_{\text{diss}}$ to the vortices which will extract energy from the small scales. Similarly to the forcing velocity in FourierBandForcingBS, such dissipation velocity can be written as:

\[\bm{v}_{\text{diss}}(\bm{s}) = -α \bm{s}' × \bm{v}^{>}(\bm{s})\]

where $\bm{v}^{>}(\bm{s})$ denotes the high-pass filtered superfluid velocity (from Biot–Savart's law) at scales $k > k_{\text{diss}}$.

In practice, this velocity is obtained as $\bm{v}^{>} = \bm{v} - \bm{v}^{<}$, where $\bm{v}$ and $\bm{v}^{<}$ are the total and low-pass filtered superfluid velocities (up to scale $k_{\text{diss}}$ included). This means that the Fourier-space fields must be sufficiently well resolved, up to the chosen dissipative scale. Evaluating $\bm{v}^{<}$ at the vortex positions requires an extra type-2 NUFFT (interpolation from Fourier to physical space), which is a bit costly.

Energy dissipation rate

At scales $k > k_{\text{diss}}$, the dissipation rate associated to this term is:

\[ε_{\text{diss}} = -α \frac{Γ}{V} ∮ |\bm{s}' × \bm{v}^{>}|^2 \, \mathrm{d}ξ\]

AbstractForcing

Abstract type representing a forcing method.

Note that, in general, all forcing methods can also be used as dissipation methods (e.g. by setting forcing coefficients to a negative value).

AbstractDissipation

Abstract type representing a dissipation method (via an extra dissipation term in the vortex velocity).

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.