Functions

Ellipsoid{T<:Number} <: Volume{T}

Volume type Ellipsoid. Create by calling Ellipsoid(a,b,c), with a,b,c the semi-axes. a,b,c can be of any Number type.

Examples: Ellipsoid(1.1,1.0,0.9) is a Ellipsoid{Float64}, Ellipsoid(1//1,1//2,1//5) is a Ellispoid{Rational{Int64}}.

Sphere{T<:Number} <: Volume{T}

Volume type Sphere. Create by calling Sphere{T}(), with T any Number type. Default: Sphere(T) gives a Sphere{Float64}(). For other types use, e.g. Sphere{Rational{BigInt}}().

LebovitzBasis{T<:Number,Vol<:Volume{T}} <: VectorBasis{T,Vol}

Basis of 3-D vector field, so that $\mathbf{u}\cdot\mathbf{n} = 0$ at $\partial\mathcal{V}$ and $\nabla\cdot\mathbf{u} = 0$ after Lebovitz (1989).

QGBasis{T<:Number,Vol<:Volume{T}} <: VectorBasis{T,Vol}

Basis of QG vector field (Gerick et al., 2020), so that $\mathbf{u}=\nabla(h^3x^ny^m)\times\nabla(z/h)$.

QGIMBasis{T<:Number,Vol<:Sphere{T}} <: VectorBasis{T,Vol}

Basis of complex QG inertial modes (Maffei et al., 2017). This basis is orthonormal.

QGRIMBasis{T<:Number,Vol<:Sphere{T}} <: VectorBasis{T,Vol}

Real basis similar to the QG inertial modes (Maffei et al., 2017). They are not the solutions to the QG inertial mode equation and they are not orthogonal.

ConductingMFBasis{T<:Number,Vol<:Volume{T}} <: VectorBasis{T,Vol}

Basis of 3-D magnetic field, so that $\mathbf{B}\cdot\mathbf{n} = 0$ at $\partial\mathcal{V}$ and $\nabla\cdot\mathbf{B} = 0$ after Lebovitz (1989). It is exactly the same as LebovitzBasis.

InsulatingMFBasis{T<:Number,Vol<:Sphere{T}} <: VectorBasis{T,Vol}

Basis of insulating magnetic fields following Gerick et al. (2021). For now, only for Vol<:Sphere{T}, i.e. in a spherical domain.

HDProblem{T<:Number,Vol<:Volume{T}} <: MireProblem{T, Vol}

Defines hydrodynamic problem.

Example:

N = 5
Ω = [0,0,1.0]
V = Ellipsoid(1.1,1.0,0.9)
problem = HDProblem(N,V,Ω,LebovitzBasis)

MHDProblem{T<:Number,Vol<:Volume{T}} <: MireProblem{T,Vol}

Defines magnetohydrodynamic problem.

Example for a hybrid QG model with 3-D magnetic field with conducting boundary condition and QG velocities:

N = 5
Ω = [0,0,1.0]
a,b,c = 1.1,1.0,0.9
V = Ellipsoid(a,b,c)
B0 = [-y/b^2,x/a^2,0] #Malkus field
problem = MHDProblem(N,V,Ω,B0,QGBasis,ConductingMFBasis)

assemble!(P::HDProblem{T,V}; threads=false, verbose=false, kwargs...) where {T,V}

Assembles the matrices P.LHS and P.RHS, i.e. projecting the velocity basis P.vbasis on the inertial acceleration and Coriolis force.

assemble!(P::MHDProblem{T,V}) where {T,V}

Assembles the matrices P.LHS and P.RHS, i.e. projecting the velocity and magnetic field bases on the inertial acceleration, Coriolis force, Lorentz force and mgnetic advection.

projectforce(vs_i, vs_j, cmat, forcefun, args...; kwargs...)

Project basis vs_i onto the forcing forcefun(vs_j, args...) using precached monomials in cmat.

projectforce!(i0, j0, itemps, jtemps, valtemps, cmat, vs_i, vs_j, forcefun, args...; verbose=false, thresh=10eps())

Project basis vs_i onto the forcing forcefun(vs_j, args...). Pushes indices and values of non-zero entries into the itemps,jtemps and valtems arrays. i0 and j0 are the origin of indices (useful for combining multiple equations, should be 0 to have no shift in indices). See projectforce to construct a sparse array from the indices and values.

projectforcet!(args...)

Multithreaded version of projectforce!. Input has to be adapted to number of threads!

inner_product(u, v, cmat)

Inner product in an ellipsoidal volume $\int \mathbf{u}\cdot \mathbf{v} dV$, using precached monomial integrations in cmat.

inner_product(u, v, a, b, c)

Inner product in an ellipsoidal volume $\int \mathbf{u}\cdot \mathbf{v} dV$.