Baroclinic instability of a 2D front based on Eady (1949)
Introduction
Eady (1949) showed that in a uniformly sheared, stratified layer between two rigid lids on an $f$-plane, two counter-propagating Rossby edge waves can phase lock and convert available potential energy into kinetic energy, producing baroclinic eddies that grow fastest at wavelengths about four deformation radii and on timescales of a few days.
Basic state
The basic state is given by
\[\begin{align} B(y, z) &= Ri z - y, \\ U(y, z) &= z - {1}/{2}, \end{align}\]
where $Ri$ is the Richardson number, and $N^2 = Ri$ is the stratification.
Governing equations
The non-dimensional form of the linearized version of the QG PV perturbation equation under the $f$-plane approximation can be expressed as,
\[\begin{align} \frac{\partial q^\text{qg}}{\partial t} + U \frac{\partial q^\text{qg}}{\partial x} + \frac{\partial \psi}{\partial x} \frac{\partial Q^\text{qg}}{\partial y} = E \, \nabla_h^2 q^\text{qg}, \,\,\,\,\,\,\ \text{for} \,\,\, 0 < z <1, \end{align}\]
where $q^\text{qg}$ is the perturbation QG PV, and it is defined as
\[\begin{align} q^\text{qg} = \nabla_h^2 \psi^\text{qg} + \frac{\partial}{\partial z} \left(\frac{1}{N^2} \frac{\partial \psi^\text{qg}}{\partial z}\right), \end{align}\]
The variable $\psi^\text{qg}$ describes the QG perturbation streamfunction with $u^\text{qg}=-\partial_y \psi^\text{qg}$ and $v^\text{qg}=\partial_x \psi^\text{qg}$. The variable $Q^\text{qg}$ describes the QG PV of the basic state, which is defined as \citep{pedlosky2013geophysical}
\[\begin{align} Q^\text{qg} = -\frac{\partial U}{\partial y} + \frac{\partial}{\partial z}\left(\frac{B}{N^2} \right), \end{align}\]
and the cross-front gradient of $Q^\text{qg}$ is defined as
\[\begin{align} \frac{\partial Q^\text{qg}}{\partial y} = - \frac{\partial}{\partial z}\left(\frac{\partial_z U}{N^2} \right). \end{align}\]
The linearized perturbation buoyancy equation at the top and the bottom boundary is
\[\begin{align} \frac{\partial b^\text{qg}}{\partial t} + U \frac{\partial b^\text{qg}}{\partial x} + \frac{\partial \psi^\text{qg}}{\partial x} \frac{\partial B}{\partial y} = 0, \,\,\,\,\,\,\ \text{at} \, z=0 \,\ \text{and} \,\, 1, \end{align}\]
where $b^\text{qg}=\partial_z \psi^\text{qg}$.
Normal-mode solutions
Next, we seek normal-mode solutions for $\psi^\text{qg}$ and $q^\text{qg}$ in the form of
\[\begin{align} [\psi^\text{qg}, q^\text{qg}] = \mathfrak{R}\big([\widetilde{\psi}^\text{qg}, \widetilde{q}^\text{qg}] \big)(y, z) e^{i kx-\sigma t}, \end{align}\]
where $\widetilde{\psi}^\text{qg}$, $\widetilde{q}^\text{qg}$ are the eigenfunctions of $\psi^\text{qg}$ and $q^\text{qg}$, respectively. In terms of streamfunction $\psi^\text{qg}$,
\[\begin{align} [(\sigma + i k U) - E] \mathscr{L}\widetilde{\psi}^\text{qg} + i k \partial_y Q^\text{qg} \widetilde{\psi}^\text{qg} &= 0, \,\,\,\,\ \text{for} \,\, 0 < z <1, \\ (\sigma + i k U_{-})\partial_z \widetilde{\psi}^\text{qg}_{-} + i k \partial_y B_{-} \widetilde{\psi}^\text{qg}_{-} &= 0, \,\,\,\,\, \text{at} \,\, z = 0, \\ (\sigma + i k U_{+})\partial_z \widetilde{\psi}^\text{qg}_{+} + i k \partial_y B_{+} \widetilde{\psi}^\text{qg}_{+} &= 0, \,\,\,\,\, \text{at} \,\, z = 1, \end{align}\]
where $\mathscr{L}$ is a linear operator, and is defined as $\mathscr{L} \equiv \mathcal{D}_h^2 + 1/N^2 \partial_z^2$, where $\mathcal{D}_h^2 = (\partial_y^2 - k^2)$. The subscripts $-,+$ denote the values of the fields at $z=0$ and $z=1$, respectively.
Generalized eigenvalue problem
The above set of equations can be cast into a generalized eigenvalue problem
\[\begin{align} AX= λBX, \end{align}\]
where $\lambda$ is the eigenvalue, and $X$ is the eigenvector. The matrices $A$ and $B$ are given by
\[\begin{align} A &= \begin{bmatrix} ik U \mathcal{D}_0^2 + ik \partial_y Q - E \mathcal{D}_h^2 \mathcal{D}_0^2 \end{bmatrix}, \,\,\,\,\,\,\, B &= \begin{bmatrix} - \mathcal{D}_0^2 \end{bmatrix}, \end{align}\]
where $D_0^2 = \mathcal{D}_h^2 + (1/N^2) \mathcal{D}_z^2$.
Load required packages
using LazyGrids
using LinearAlgebra
using Printf
using StaticArrays
using SparseArrays
using SparseMatrixDicts
using FillArrays
using SpecialFunctions
using Parameters
using Test
using BenchmarkTools
using JLD2
using Parameters: @with_kw
using BiGSTARS
using BiGSTARS: AbstractParams
using BiGSTARS: Problem, OperatorI, TwoDGridParameters
@with_kw mutable struct Params{T} <: AbstractParams
L::T = 1.0 # horizontal domain size
H::T = 1.0 # vertical domain size
Ri::T = 1.0 # the Richardson number
k::T = 0.1 # along-front wavenumber
E::T = 1.0e-12 # the Ekman number
Ny::Int64 = 60 # no. of y-grid points
Nz::Int64 = 30 # no. of z-grid points (should be different from Ny)
eig_solver::String = "arpack" # eigenvalue solver
endBasic state
function basic_state(grid, params)
Y, Z = ndgrid(grid.y, grid.z)
Y = transpose(Y)
Z = transpose(Z)
# Define the basic state
B = @. 1.0 * params.Ri * Z - Y # buoyancy
U = @. 1.0 * Z - 0.5 * params.H # along-front velocity
# Calculate all the 1st, 2nd and yz derivatives in 2D grids
bs = compute_derivatives(U, B, grid.y; grid.Dᶻ, grid.D²ᶻ, gridtype = :All)
precompute!(bs; which = :All) # eager cache, returns bs itself
@assert bs.U === U # originals live in the same object
@assert bs.B === B
return bs
endConstructing Generalized EVP
function generalized_EigValProb(prob, grid, params)
# basic state
bs = basic_state(grid, params)
N = params.Ny * params.Nz
I⁰ = sparse(Matrix(1.0I, N, N)) # Identity matrix
s₁ = size(I⁰, 1);
s₂ = size(I⁰, 2);
# the horizontal Laplacian operator: ∇ₕ² = ∂ʸʸ - k²
∇ₕ² = SparseMatrixCSC(Zeros(N, N))
∇ₕ² = (1.0 * prob.D²ʸ - 1.0 * params.k^2 * I⁰)
# some quantities required later
∂ᶻB⁻¹ = @. 1.0/bs.∂ᶻB
∂ᶻB⁻² = @. 1.0/(bs.∂ᶻB * bs.∂ᶻB)
∂ʸQ::Array{Float64, 2} = SparseMatrixCSC(Zeros(N, N)) # PV gradient is zero
# definition of perturbation PV, q = D₂³ᵈ{ψ}
D₂³ᵈ = (1.0 * ∇ₕ²
+ 1.0 * DiagM(∂ᶻB⁻¹) * prob.D²ᶻ
- 1.0 * DiagM(bs.∂ᶻᶻB) * DiagM(∂ᶻB⁻²) * prob.Dᶻ)
# Construct the matrix `A`
# ──────────────────────────────────────────────────────────────────────────────
# 1) Now define your 1×1 block-rows in a NamedTuple of 1-tuples
# ──────────────────────────────────────────────────────────────────────────────
# Construct the matrix `A`
Ablocks = (
ψ = ( # ψ-equation
sparse(1.0im * params.k * DiagM(bs.U) * D₂³ᵈ
+ 1.0im * params.k * ∂ʸQ
- 1.0 * params.E * ∇ₕ² * D₂³ᵈ
)
),
)
# Construct the matrix `B`
Bblocks = (
ψ = ( # ψ-equation: [-D₂³ᵈ]
sparse(-D₂³ᵈ)
),
)
# ──────────────────────────────────────────────────────────────────────────────
# 2) Assemble the block-row matrices into a GEVPMatrices object
# ──────────────────────────────────────────────────────────────────────────────
gevp = GEVPMatrices(Ablocks, Bblocks)
# ──────────────────────────────────────────────────────────────────────────────
# 3) And now you have exactly:
# gevp.A, gevp.B → full sparse matrices
# gevp.As.w, gevp.As.ζ, gevp.As.b → each block-row view
# gevp.Bs.w, gevp.Bs.ζ, gevp.Bs.b
# ──────────────────────────────────────────────────────────────────────────────
B = SparseMatrixCSC(Zeros{ComplexF64}(s₁, s₂))
C = SparseMatrixCSC(Zeros{ Float64 }(s₁, s₂))
# ──────────────────────────────────────────────────────────────────────────────
# 4) Implementing boundary conditions
# ──────────────────────────────────────────────────────────────────────────────
_, zi = ndgrid(1:1:params.Ny, 1:1:params.Nz)
zi = transpose(zi);
zi = zi[:];
bcᶻ⁻ = findall( x -> (x==1), zi ) ## @ z=0
bcᶻ⁺ = findall( x -> (x==params.Nz), zi ) ## @ z=1
# Implementing boundary condition for 𝓛 matrix in the z-direction:
B[:,1:1s₂] = 1.0im * params.k * DiagM(bs.U) * prob.Dᶻ - 1.0im * params.k * DiagM(bs.∂ᶻU)
# Bottom boundary condition @ z=0
@. gevp.A[bcᶻ⁻, :] = B[bcᶻ⁻, :]
# Top boundary condition @ z = 1
@. gevp.A[bcᶻ⁺, :] = B[bcᶻ⁺, :]
# Implementing boundary condition for ℳ matrix in the z-direction:
C[:,1:1s₂] = -1.0 * prob.Dᶻ
# Bottom boundary condition @ z=0
@. gevp.B[bcᶻ⁻, :] = C[bcᶻ⁻, :]
# Top boundary condition @ z = 1
@. gevp.B[bcᶻ⁺, :] = C[bcᶻ⁺, :]
return gevp.A, gevp.B
endEigenvalue solver
function EigSolver(prob, grid, params, σ₀)
A, B = generalized_EigValProb(prob, grid, params)
# Construct the eigenvalue solver
# Methods available: :Krylov (by default), :Arnoldi, :Arpack
# Here we are looking for largest growth rate (real part of eigenvalue)
solver = EigenSolver(A, B; σ₀=σ₀, method=:Krylov, nev=1, which=:LR, sortby=:R)
solve!(solver)
λ, Χ = get_results(solver)
print_summary(solver)
# Print the largest growth rate
@printf "largest growth rate : %1.4e%+1.4eim\n" real(λ[1]) imag(λ[1])
return λ[1], Χ[:,1]
endSolving the Eady problem
function solve_Eady(k::Float64)
# Calling problem parameters
params = Params{Float64}()
# Construct grid and derivative operators
grid = TwoDGrid(params)
# Construct the necesary operator
ops = OperatorI(params)
prob = Problem(grid, ops)
# update the wavenumber
params.k = k
# initial guess for the growth rate
σ₀ = 0.02
λ, Χ = EigSolver(prob, grid, params, σ₀)
# Analytical solution of Eady (1949) for the growth rate
μ = 1.0 * params.k * √params.Ri
λₜ = 1.0/√params.Ri * √( (coth(0.5μ) - 0.5μ)*(0.5μ - tanh(0.5μ)) )
@printf "Analytical solution of Eady (1949) for the growth rate: %f \n" λₜ
return abs(λ.re - λₜ) < 1e-3
endResult
solve_Eady(0.1) # growth rate is at k=0.1(attempt 1/16) trying σ = 0.024000 with Krylov
✓ converged: λ₁ = 0.028829 + -0.000000i
(attempt 2/16) trying σ = 0.024800 with Krylov
✓ converged: λ₁ = 0.028829 + 0.000000i
✓ successive eigenvalues converged: |Δλ| = 4.77e-13 < 1.00e-05
EigenSolver Results Summary
Method: Krylov
Converged: ✅ Yes
Final shift: 0.0248
Total time: 3.773s
Attempts: 2
├─ Eigenvalues (1) ───────────────
│ i │ λ (Re Im·i) │
│───┼──────────────────────│
│ 1 │ 0.028829 +0.000000i │
└──────────────────────────┘
largest growth rate : 2.8829e-02+2.3252e-13im
Analytical solution of Eady (1949) for the growth rate: 0.028829
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