Functions

DefLQGParams(; U=1, ℓ=1, R=Inf, β=0, ActiveLayers=1, H=1, x₀=[0,0], α=0, M=8, tol=1e-6, K₀=nothing, a₀=nothing, UseAnalytic=false, CalcVelocity=false, CalcVorticity=false, CalcEnergy=false, CalcEnstrophy=false)

Defines an LQGParams structure using the given inputs

Keyword arguments:

U: Vortex speed
ℓ: Vortex radius
R: Rossby radius
β: background PV gradient
ActiveLayers: 1 => layer contains vortex region
H: thickness of each layer
x₀: Vortex position
α: Direction of vortex propagation
M: number of coefficients in Zernike expansion
tol: maximum error in solution evaluation
K₀: initial guess for eigenvalue
a₀: initial guess for coefficients
UseAnalytic: use analytic solution (1-layer only)
CalcVelocity: flag to determine if velocity is calculated
CalcVorticity: flag to determine if vorticity is calculated
CalcEnergy: flag to determine if energy is calculated
CalcEnstrophy: flag to determine if enstrophy is calculated

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QGDipoles.DefLQGVortex — Function

DefLQGVortex(grid; U=1, ℓ=1, R=Inf, β=0, ActiveLayers=1, H=1, x₀=[0,0], α=0, M=8, tol=1e-6, K₀=nothing, a₀=nothing, UseAnalytic=false, CalcVelocity=false, CalcVorticity=false, CalcEnergy=false, CalcEnstrophy=false)

Defines an LQGVortex solution structure using the given inputs

Arguments:

grid: grid structure

Keyword arguments:

U: Vortex speed
ℓ: Vortex radius
R: Rossby radius
β: background PV gradient
ActiveLayers: 1 => layer contains vortex region
H: thickness of each layer
x₀: Vortex position
α: Direction of vortex propagation
M: number of coefficients in Zernike expansion
tol: maximum error in solution evaluation
K₀: initial guess for eigenvalue
a₀: initial guess for coefficients
UseAnalytic: use analytic solution (1-layer only)
CalcVelocity: flag to determine if velocity is calculated
CalcVorticity: flag to determine if vorticity is calculated
CalcEnergy: flag to determine if energy is calculated
CalcEnstrophy: flag to determine if enstrophy is calculated

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DefLQGVortex(grid, params)

Defines an LQGVortex solution structure using the given inputs

Arguments:

grid: grid structure
params: vortex parameters, LQGParams structure

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SQG

QGDipoles.SQGParams — Type

SQGParams

Stores the parameters for an SQG dipolar vortex solution

Fields:

U: Vortex speed
ℓ: Vortex radius
R: Rossby radius
β: background PV gradient
x₀: Vortex position
α: Direction of vortex propagation
M: number of coefficients in Zernike expansion
tol: maximum error in solution evaluation
K₀: initial guess for eigenvalue
a₀: initial guess for coefficients
CalcVelocity: flag to determine if velocity is calculated
CalcVorticity: flag to determine if vorticity is calculated
CalcEnergy: flag to determine if energy is calculated

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QGDipoles.SQGVortex — Type

SQGVortex

Stores fields and diagnostics for an SQG dipolar vortex solution

Fields:

params: Vortex params
ψ: surface streamfunction
b: surface buoyancy
K: eigenvalue
a: coefficient matrix
u: x velocity
v: y velocity
ζ: vertical vorticity
E: domain integrated energy
SPE: surface potential energy

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QGDipoles.DefSQGParams — Function

DefSQGParams(; U=1, ℓ=1, R=[Inf,Inf], β=0, x₀=[0,0], α=0, M=12, tol=1e-6, K₀=nothing, a₀=nothing, CalcVelocity=false, CalcVorticity=false, CalcEnergy=false)

Defines an SQGParams structure using the given inputs

Keyword arguments:

U: Vortex speed
ℓ: Vortex radius
R: Rossby radius
β: background PV gradient
x₀: Vortex position
α: Direction of vortex propagation
M: number of coefficients in Zernike expansion
tol: maximum error in solution evaluation
K₀: initial guess for eigenvalue
a₀: initial guess for coefficients
CalcVelocity: flag to determine if velocity is calculated
CalcVorticity: flag to determine if vorticity is calculated
CalcEnergy: flag to determine if energy is calculated

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QGDipoles.DefSQGVortex — Function

DefSQGVortex(grid; U=1, ℓ=1, R=[Inf,Inf], β=0, x₀=[0,0], α=0, M=12, tol=1e-6, K₀=nothing, a₀=nothing, CalcVelocity=false, CalcVorticity=false, CalcEnergy=false)

Defines an SQGVortex solution structure using the given inputs

Arguments:

grid: grid structure

Keyword arguments:

U: Vortex speed
ℓ: Vortex radius
R: Rossby radius
β: background PV gradient
x₀: Vortex position
α: Direction of vortex propagation
M: number of coefficients in Zernike expansion
tol: maximum error in solution evaluation
K₀: initial guess for eigenvalue
a₀: initial guess for coefficients
CalcVelocity: flag to determine if velocity is calculated
CalcVorticity: flag to determine if vorticity is calculated
CalcEnergy: flag to determine if energy is calculated

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DefSQGVortex(grid, params)

Defines an SQGVortex solution structure using the given inputs

Arguments:

grid: grid structure
params: vortex parameters, LQGParams structure

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Shared

QGDipoles.UpdateParams — Function

UpdateParams(params::LQGParams; kwargs...)

Creates an LQGParams structure by replacing parameters in params with the given keywords

Arguments:

params: LQGParams parameter structure

Keyword arguments:

kwargs...: keyword arguments for DefLQGParams

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UpdateParams(params::SQGParams; kwargs...)

Creates an SQGParams structure by replacing parameters in params with the given keywords

Arguments:

params: SQGParams parameter structure

Keyword arguments:

kwargs...: keyword arguments for DefSQGParams

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QGDipoles.UpdateVortex — Function

UpdateVortex(grid, vortex::LQGVortex; kwargs...)

Creates an LQGVortex structure by replacing parameters in vortex.params with the given keywords

Arguments:

grid: grid structure
vortex: LQGVortex structure

Keyword arguments:

kwargs...: keyword arguments for DefLQGParams

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UpdateVortex(grid, vortex::SQGVortex; kwargs...)

Creates an SQGVortex structure by replacing parameters in vortex.params with the given keywords

Arguments:

grid: grid structure
vortex: SQGVortex structure

Keyword arguments:

kwargs...: keyword arguments for DefSQGParams

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High-Level Functions

LQG

QGDipoles.CreateModonLQG — Function

CreateModonLQG(grid; U=1, ℓ=1, R=Inf, β=0, ActiveLayers=1, x₀=[0, 0], α=0, M=8, tol=1e-6, K₀=nothing, a₀=nothing)

High level wrapper function for calculating $ψ$ and $q$ for the Layered QG model using given parameters

Arguments:

grid: grid structure containing x, y, and Krsq

Keyword arguments:

(U, ℓ): vortex speed and radius, Numbers (default: (1, 1))
(R, β): Rossby radii and (y) PV gradients in each layer, Numbers or Vectors, (default: (Inf, 0))
ActiveLayers: vector of 1s or 0s where 1 denotes an active layer, Number or Vector, (default: [1,..,1])
x₀: position of vortex center, vector (default: [0, 0])
α: initial angle of vortex, Number (default: 0)
M: number of coefficient to solve for, Integer (default: 8)
tol: error tolerance passed to QuadGK and NLSolve functions, Number (default: 1e-6)
K₀, a₀: initial guesses for $K$ and $a$, Arrays or nothings (default: nothing)

Note: provide values of K₀ and a₀ for active layers ONLY.

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QGDipoles.CreateLCD — Function

CreateLCD(grid; U=1, ℓ=1, x₀=[0, 0], α=0)

High level wrapper function for calculating $ψ$ and $q$ for the Lamb-Chaplygin dipole using given parameters

Arguments:

grid: grid structure containing x, y, and Krsq

Keyword arguments:

(U, ℓ): vortex speed and radius, Numbers (default: (1, 1))
x₀: position of vortex center, vector (default: [0, 0])
α: initial angle of vortex, Number (default: 0)

Note: This function uses the analytic solution for the LCD to calculate $ψ$ and $q$.

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QGDipoles.CreateLRD — Function

CreateLRD(grid; U=1, ℓ=1, R=Inf, β=0, x₀=[0, 0], α=0)

High level wrapper function for calculating $ψ$ and $q$ for the Larichev-Reznik dipole using given parameters

Arguments:

grid: grid structure containing x, y, and Krsq

Keyword arguments:

(U, ℓ): vortex speed and radius, Numbers (default: (1, 1))
(R, β): Rossby radii and (y) PV gradient, Numbers, (default: (Inf, 0))
x₀: position of vortex center, vector (default: [0, 0])
α: initial angle of vortex, Number (default: 0)

Note: This function uses the analytic solution for the LRD to calculate $ψ$ and $q$.

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SQG

QGDipoles.CreateModonSQG — Function

CreateModonSQG(grid; U=1, ℓ=1, R=[Inf, Inf], β=0, x₀=[0, 0], α=0, M=12, tol=1e-6, K₀=nothing, a₀=nothing)

High level wrapper function for calculating $ψ$ and $b$ for the SQG model using given parameters

Arguments:

grid: grid structure containing x, y, and Krsq

Keyword arguments:

(U, ℓ): vortex speed and radius, Numbers (default: (1, 1))
R: vector of $[R, R']$, Vector (default: [Inf, Inf])
β: beta-plane (y) PV gradient, Number (default: 0)
x₀: position of vortex center, vector (default: [0, 0])
α: initial angle of vortex, Number (default: 0)
M: number of coefficient to solve for, Integer (default: 12)
tol: error tolerance passed to QuadGK and NLSolve functions, Number (default: 1e-6)
K₀, a₀: initial guesses for $K$ and $a$, Arrays or nothings (default: nothing)

Note: Here R is the baroclinic Rossby radius, R = NH/f, and R' = R₀²/R where R₀ is the barotropic Rossby radius, R₀ = √(gH)/f. For infinite depth, R' = g/(fN).

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QGDipoles.Eval_ψ_SQG — Function

Eval_ψ_SQG(grid, ψ; z=[0], U=1, R=[Inf, Inf], β=0)

Evaluates $ψ$ at specified depths, $z ∈ [-R, 0]$, for the SQG problem

Arguments:

grid: grid structure containing x, y, and Krsq
ψ: surface streamfunction, calculated using Calc_ψb or CreateModonSQG

Keyword arguments:

z: vector of depths (default: [0])
U: vortex speed, Number (default: 1)
R: vector of $[R, R']$, Vector (default: [Inf, Inf])
β: beta-plane (y) PV gradient, Number (default: 0)

Note: Here $R$ is the baroclinic Rossby radius, $R = NH/f$, and $R' = R₀²/R$ where $R₀$ is the barotropic Rossby radius, $R₀ = √(gH)/f$. For infinite depth, $R' = g/(fN)$.

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QGDipoles.Eval_q_SQG — Function

Eval_q_SQG(grid, ψ; z=[0], U=1, R=[Inf, Inf], β=0)

Evaluates $q$ at specified depths, $z ∈ [-R, 0]$, for the SQG problem

Arguments:

grid: grid structure containing x, y, and Krsq
ψ: surface streamfunction, calculated using Calc_ψb or CreateModonSQG

Keyword arguments:

z: vector of depths (default: [0])
U: vortex speed, Number (default: 1)
R: vector of $[R, R']$, Vector (default: [Inf, Inf])
β: beta-plane (y) PV gradient, Number (default: 0)

Note: Here $R$ is the baroclinic Rossby radius, $R = NH/f$, and $R' = R₀²/R$ where $R₀$ is the barotropic Rossby radius, $R₀ = √(gH)/f$. For infinite depth, $R' = g/(fN)$.

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QGDipoles.Eval_b_SQG — Function

Eval_b_SQG(grid, ψ; z=[0], U=1, R=[Inf, Inf], β=0)

Evaluates $b$ at specified depths, $z ∈ [-R, 0]$, for the SQG problem

Arguments:

grid: grid structure containing x, y, and Krsq
ψ: surface streamfunction, calculated using Calc_ψb or CreateModonSQG

Keyword arguments:

z: vector of depths (default: [0])
U: vortex speed, Number (default: 1)
R: vector of $[R, R']$, Vector (default: [Inf, Inf])
β: beta-plane (y) PV gradient, Number (default: 0)

Note: Here $R$ is the baroclinic Rossby radius, $R = NH/f$, and $R' = R₀²/R$ where $R₀$ is the barotropic Rossby radius, $R₀ = √(gH)/f$. For infinite depth, $R' = g/(fN)$.

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QGDipoles.Eval_w_SQG — Function

Eval_w_SQG(grid, ψ; z=[0], U=1, R=[Inf, Inf], β=0)

Evaluates N²w at specified depths, $z ∈ [-R, 0]$, for the SQG problem using $N²w = -J[ψ + Uy, b]$

Arguments:

grid: grid structure containing x, y, and Krsq
ψ: surface streamfunction, calculated using Calc_ψb or CreateModonSQG

Keyword arguments:

z: vector of depths (default: [0])
U: vortex speed, Number (default: 1)
R: vector of $[R, R']$, Vector (default: [Inf, Inf])
β: beta-plane (y) PV gradient, Number (default: 0)

Note: Here $R$ is the baroclinic Rossby radius, $R = NH/f$, and $R' = R₀²/R$ where $R₀$ is the barotropic Rossby radius, $R₀ = √(gH)/f$. For infinite depth, $R' = g/(fN)$.

Note: this function is not accurate at the surface as $∇b$ is discontinuous there. Instead use $w = -U∂η/∂x$ where $η = fψ/g$ is the surface elevation, or $w = 0$ if $R' = ∞$.

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Utilities

QGDipoles.CreateGrid — Function

CreateGrid(Nx, Ny, Lx, Ly; cuda=false)

Define the numerical grid as a GridStruct

Arguments:

Nx, Ny: number of gridpoints in x and y directions, Integers
Lx, Ly: x and y domains, either vectors of endpoints or lengths, Vectors or Numbers

Keyword arguments:

cuda: true; use CUDA CuArray for fields (default: false)

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CreateGrid(; Nx=512, Ny=512, Lx=[-5,5], Ly=[-5,5], cuda=false)

Define the numerical grid as a GridStruct using a keyword-based method

Keyword arguments:

Nx, Ny: number of gridpoints in x and y directions, Integers
Lx, Ly: x and y domains, either vectors of endpoints or lengths, Vectors or Numbers
cuda: true; use CUDA CuArray for fields (default: false)

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QGDipoles.Calc_uv — Function

Calc_uv(grid, ψ)

Calculate the velocity fields from $ψ$ using $(u, v) = (-∂ψ/∂y, ∂ψ/∂x)$

Arguments:

grid: grid structure containing kr and l
ψ: streamfunction, Array

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QGDipoles.Calc_∇ — Function

Calc_∇(grid, f)

Calculate the gradient $∇f$ for a given field $f$

Arguments:

grid: grid structure containing kr and l
f: function, Array

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QGDipoles.Calc_ζ — Function

Calc_ζ(grid, ψ)

Calculate the vertical vorticity using $ζ = ∂v/∂x - ∂u/∂y = ∇²ψ$

Arguments:

grid: grid structure containing Krsq
ψ: streamfunction, Array

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Diagnostics

LQG

QGDipoles.EnergyLQG — Function

EnergyLQG(grid, ψ; R=Inf, H=[1])

Calculates the kinetic and potential energy for the LQG system

Arguments:

grid: grid structure containing Krsq
ψ: streamfunction in each layer, Array or CuArray

Keyword arguments:

R: Rossby radius in each layer, Number or Vector (default: Inf)
H: Thickness of each layer, Number or Vector (default: [1])

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QGDipoles.EnstrophyLQG — Function

EnstrophyLQG(grid, q; H=[1])

Calculates the enstrophy for the LQG system

Arguments:

grid: grid structure containing Krsq
q: potential vorticity anomaly in each layer, Array or CuArray

Keyword arguments:

H: Thickness of each layer, Number or Vector (default: [1])

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SQG

QGDipoles.EnergySQG — Function

EnergySQG(grid, ψ, b; R′)

Calculates the energies for the SQG system; the total domain integrated energy and the surface potential energy

Arguments:

grid: grid structure containing Krsq
ψ: surface streamfunction, Array or CuArray
b: surface buoyancy, , Array or CuArray

Keyword arguments:

R′: reduced barotropic Rossby radius, Number (default: Inf)

Note: the surface potential energy is sometimes referred to as the generalised enstrophy or the buoyancy variance.

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Low-Level Functions

LQG

QGDipoles.BuildLinSysLQG — Function

BuildLinSysLQG(M, λ, μ; tol=1e-6)

Builds the terms in the inhomogeneous eigenvalue problem; $A$, $B$, $c$, $d$ for the LQG problem

Arguments:

M: number of coefficient to solve for, Integer
λ: ratio of vortex radius to Rossby radius in each layer, Number or Vector
μ: nondimensional (y) vorticity gradient in each layer, Number or Vector

Keyword arguments:

tol: error tolerance for QuadGK via JJ_int, Number (default: 1e-6)

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QGDipoles.ApplyPassiveLayers — Function

ApplyPassiveLayers(A, B, c, d, ActiveLayers)

Removes rows and columns corresponding to passive layers from the system

Arguments:

A, B, c, d: inhomogeneous eigenvalue problem terms, Arrays
ActiveLayers: vector of 1s or 0s where 1 denotes an active layer, Number or Vector

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QGDipoles.IncludePassiveLayers — Function

IncludePassiveLayers(K, a, ActiveLayers)

Includes columns corresponding to passive layers in the eigenvalue and coefficient arrays

Arguments:

K, a: eigenvalue and coefficient arrays describing system solution, Arrays
ActiveLayers: vector of 1s or 0s where 1 denotes an active layer, Number or Vector

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QGDipoles.Calc_ψq — Function

Calc_ψq(grid, a; U, ℓ, R, β, x₀=[0, 0], α=0)

Calculate $ψ$ and $q$ in a layered QG model using coefficients and vortex parameters

Arguments:

grid: grid structure containing x, y, and Krsq
a: M x N array of coefficients, Array

Keyword arguments:

(U, ℓ): vortex speed and radius, Numbers (default: (1, 1))
(R, β): Rossby radii and (y) PV gradients in each layer, Numbers or Vectors (default: (Inf, 0))
x₀: position of vortex center, vector (default: [0, 0])
α: initial angle of vortex, Number (default: 0)

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SQG

QGDipoles.BuildLinSysSQG — Function

BuildLinSysSQG(M, λ, μ; tol=1e-6)

Builds the terms in the inhomogeneous eigenvalue problem; $A$, $B$, $c$, $d$ for the SQG problem

Arguments:

M: number of coefficient to solve for, Integer
λ: ratio of vortex radius to Rossby radius in each layer, Number or Vector
μ: nondimensional (y) vorticity gradient in each layer, Number or Vector

Keyword arguments:

tol: error tolerance for QuadGK via JJ_int, Number (default: 1e-6)

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QGDipoles.Calc_ψb — Function

Calc_ψb(grid, a; U, ℓ, R, β, x₀=[0, 0], α=0)

Calculate SQG fields $ψ$ and $b$ using coefficients and vortex parameters

Arguments:

grid: grid structure containing x, y, and Krsq
a: M x 1 array of coefficients, Array

Keyword arguments:

(U, ℓ): vortex speed and radius, Numbers (default: 1)
R: vector of $[R, R']$, Vector (default: [Inf, Inf])
β: beta-plane (y) PV gradient, Number (default: 1)
x₀: position of vortex center, vector (default: [0, 0])
α: initial angle of vortex, Number (default: 0)

Note: Here R is the baroclinic Rossby radius, R = NH/f, and R' = R₀²/R where R₀ is the barotropic Rossby radius, R₀ = √(gH)/f. For infinite depth, R' = g/(fN).

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Shared

QGDipoles.SolveInhomEVP — Function

SolveInhomEVP(A, B, c, d; K₀=nothing, a₀=nothing, tol=1e-6, method=:eigensolve, m=2, warn=true)

Solves the inhomogeneous eigenvalue problem using nonlinear root finding

Arguments:

A, B, c, d: inhomogeneous eigenvalue problem terms, Arrays

Keyword arguments:

K₀, a₀: initial guesses for $K$ and $a$, Arrays or nothings (default: nothing)
tol: error tolerance for nlsolve, Number (default: 1e-6)
method: :eigensolve or :nlsolve, for $N > 1$ :nlsolve is used automatically (default: :eigensolve)
m: exponent of $K$ in eignevalue problem (default: 2)
warn: if true displays warning if solution includes unextracted passive layers (default: true)

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Extras

Monopoles

QGDipoles.CreateRankine — Function

CreateRankine(grid; ℓ=1, Γ=2π, x₀=[0, 0])

Calculates the Rankine vortex for a 1 layer system. This vortex appears as a point vortex in the far field but consists of solid body rotation within the region $r < ℓ$.

Arguments:

grid: grid structure containing x, y, and Krsq

Keyword arguments:

ℓ: vortex speed and radius, Numbers (default: 1)
Γ: vortex circulation (default: 2π)
x₀: position of vortex center, vector (default: [0, 0])

Note: This function outputs $(u, v)$ directly since the solution has discontinuous velocity at the vortex boundary, $r = ℓ$, so derivatives evaluated with Fourier transforms exhibit Gibbs phenomenon.

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QGDipoles.Create1LMonopole — Function

Create1LMonopole(grid; ℓ=1, Γ=2π, R=Inf, x₀=[0, 0])

Calculates a monopolar vortex satisfying a Long's model assumption $q = F(ψ)$ where $q = [∇²-1/R²]ψ$. We take $F(z) = -(K²+1/R²)(z-z₀)$ for $r < ℓ$ and $F(z) = 0$ for $r > ℓ$ and $z₀ = ψ(r=ℓ)$. These solutions exist only on an f-plane $(β = 0)$.

Arguments:

grid: grid structure containing x, y, and Krsq

Keyword arguments:

ℓ: vortex speed and radius, Numbers (default: 1)
Γ: vortex circulation (default: 2π)
R: Rossby radius (default: Inf)
x₀: position of vortex center, vector (default: [0, 0])

Note: This vortex has a continuous vorticity distribution so calculating (u, v) from ψ with Fourier transforms will work. This function outputs (u, v) from the analytical expressions for consistency with CreateRankine.

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QGDipoles.CreateLQGMonopole — Function

CreateLQGMonopole(grid; ℓ=1, E=1, R=Inf, x₀=[0, 0])

Calculates a monopolar vortex in the LQG model using a numerical approach. We assume that $qⱼ + βⱼ = Fⱼ(ψⱼ + Uy)$ and write $Fⱼ(z) = -Kⱼ² z + Eⱼ$. Expanding the expression gives $qⱼ + βⱼ = -Kⱼ²(ψⱼ + Uy) + Eⱼ$ which by linearity can be split into a dipole equation $qⱼ + βⱼ = -Kⱼ²(ψⱼ + Uy)$ and a monopole equation $qⱼ = Eⱼ$. Outside the vortex, we take $qⱼ = 0$.

Arguments:

grid: grid structure containing x, y, and Krsq

Keyword arguments:

ℓ: vortex speed and radius, Numbers (default: 1)
E: vector of $Eⱼ$ values, Number or Vector (default: [1, ... , 1])
R: Rossby radius (default: Inf)
x₀: position of vortex center, vector (default: [0, 0])

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QGDipoles.InvertVorticity1LQG — Function

InvertVorticity1LQG(grid, q; R=Inf)

This function inverts the potential vorticity relation $q = [∇²-1/R²]ψ$ for 1-layer QG

Arguments:

grid: grid structure containing x, y, and Krsq
q: potential vorticity field, Array

Keyword arguments:

R: Rossby radius (default: Inf)

Note: This function is designed to be used for creating periodic streamfunctions using the vorticity fields generated by Create1LMonopole. It does not support multi-layer QG and is only valid on an f-plane (β = 0).

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QGDipoles.InvertVorticityLQG — Function

InvertVorticityLQG(grid, q; R=Inf)

This function inverts the potential vorticity relation $q = ΔN ψ$ for the LQG model

Arguments:

grid: grid structure containing x, y, and Krsq
q: potential vorticity field, Array

Keyword arguments:

R: Rossby radius, Number or Vector (default: Inf)

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Internal

QGDipoles.A_func — Function

A_func(ξ, λ, μ)

Evaluates the matrix function $A(ξ, λ, μ) = K(ξ) [K(ξ) + D(μ)]⁻¹ ξ⁻¹$

Arguments:

ξ: point in $[0, ∞)$, Number
λ: ratio of vortex radius to Rossby radius in each layer, Number or Vector
μ: nondimensional (y) vorticity gradient in each layer, Number or Vector

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QGDipoles.B_func — Function

B_func(ξ, λ, μ)

Evaluates the matrix function $B(ξ, λ, μ) = [K(ξ) + D(μ)]⁻¹ ξ⁻¹$

Arguments:

ξ: point in $[0, ∞)$, Number
λ: ratio of vortex radius to Rossby radius in each layer, Number or Vector
μ: nondimensional (y) vorticity gradient in each layer, Number or Vector

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QGDipoles.JJ_int — Function

JJ_int(F, j, k, tol=1e-6)

Evaluates the integral $I = ∫_0^∞ F(ξ) J_{2j+2}(ξ) J_{2k+2}(ξ) \mathrm{d}ξ$

Arguments:

F: function to integrate, typically A_func or B_func, Function
j: first Bessel function index, Integer
k: second Bessel function index, Integer
tol: error tolerance for QuadGK, Number (default: 1e-6)

Note: This integral is performed by deforming the contour of integration into the complex plane where the Bessel function decays exponentially in the imaginary direction.

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QGDipoles.InhomEVP_F! — Function

InhomEVP_F!(F, J, x, A, B, c, d, e)

Calculates the function $F$ and it's derivatives, $J$, at a given point $x$

Arguments:

F, J: values of $F$ and it's derivatives, updated by function
x: evaluation point, Array
A, B, c: inhomogeneous eigenvalue problem terms, Arrays
e: basis spanning the space perpendicular to the $d[n]$, Array

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QGDipoles.OrthogSpace — Function

OrthogSpace(v)

Extends the input to an orthonormal basis over $R^n$ using the Gram-Schmidt method

Arguments:

v: array with vectors as columns, Array

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QGDipoles.ZernikeR — Function

ZernikeR(n, x)

Define the Zernike radial function using the jacobi function from SpecialFunctions

Arguments:

n: order, Integer
x: evaluation point, Number or Array

Note: this function is defined on $[-1, 1]$ and is set to $0$ for $|x| > 1$

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QGDipoles.GridStruct — Type

GridStruct

Stores the grid variables in physical and Fourier space

Fields:

x, y: x and y points in physical space, Ranges
kr, l: x and y points in Fourier space, Arrays
Krsq: kr²+l² in Fourier space, Array

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QGDipoles.ΔNCalc — Function

ΔNCalc(K², R, β, U=1)

Defines the $Δ_N(β)$ matrix used to invert for $ψ$ and $q$

Arguments:

K²: value of $k²+l²$ in Fourier space, Array
(R, β): Rossby radii and (y) PV gradients in each layer, Numbers or Vectors
U: vortex speed, Number (default: 1)

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QGDipoles.CartesianGrid — Function

CartesianGrid(grid)

Formats the $(x, y)$ ranges from grid as two-dimensional Arrays

Arguments:

grid: grid structure containing kr and l

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QGDipoles.PolarGrid — Function

PolarGrid(x, y, x₀)

Calculates the polar coordinates from (x, y) as two-dimensional Array centred on x₀

Arguments:

x, y: 2D Arrays for $x$ and $y$, created using CartesianGrid
x₀: Vector

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QGDipoles.AreaInteg2 — Function

AreaInteg2(f, grid)

Calculates the integral $I = ∫_A f^2 \mathrm{d}A$ where $A$ is the 2D domain described by grid.

Arguments:

f: input Array in real or Fourier space
grid: grid structure

Note: f can be entered in real space or Fourier space, we use the rfft function to calculate the Fourier transform so array sizes can distinguish the two.

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