This module solves the layered quasi-geostrophic equations on a beta-plane of variable fluid
depth H - h(x, y). The flow in each layer is obtained through a streamfunction \psi_j as
(u_j, \upsilon_j) = (-\partial_y \psi_j, \partial_x \psi_j), j = 1, \dots, n, where n
is the number of fluid layers.
The QGPV in each layer is
$$\mathrm{QGPV}_j = q_j + \underbrace{f_0 + \beta y}_{\textrm{planetary PV}} + \delta_{j,n} \underbrace{\frac{f_0 h}{H_n}}_{\textrm{topographic PV}}, \quad j = 1, \dots, n \ .$$
where q_j incorporates the relative vorticity in each layer \nabla^2\psi_j and the
vortex stretching terms:
$$q_1 = \nabla^2 \psi_1 + F_{3/2, 1} (\psi_2 - \psi_1) \ ,\\\
q_j = \nabla^2 \psi_j + F_{j-1/2, j} (\psi_{j-1} - \psi_j) + F_{j+1/2, j} (\psi_{j+1} - \psi_j) \ , \quad j = 2, \dots, n-1 \ ,\\\
q_n = \nabla^2 \psi_n + F_{n-1/2, n} (\psi_{n-1} - \psi_n) \ .$$
with
$$F_{j+1/2, k} = \frac{f_0^2}{g'_{j+1/2} H_k} \quad \text{and} \quad
g'_{j+1/2} = g \frac{\rho_{j+1} - \rho_j}{\rho_{j+1}} .$$
In view of the relationships above, when we convert to Fourier space q's and \psi's are
related via the matrix equation
$$\begin{pmatrix} \widehat{q}_{\boldsymbol{k}, 1}\\\vdots\\\widehat{q}_{\boldsymbol{k}, n} \end{pmatrix} =
\underbrace{\left(-|\boldsymbol{k}|^2 \mathbb{1} + \mathbb{F} \right)}_{\equiv \mathbb{S}_{\boldsymbol{k}}}
\begin{pmatrix} \widehat{\psi}_{\boldsymbol{k}, 1}\\\vdots\\\widehat{\psi}_{\boldsymbol{k}, n} \end{pmatrix}$$
where
$$\mathbb{F} \equiv \begin{pmatrix}
-F_{3/2, 1} & F_{3/2, 1} & 0 & \cdots & 0\\\
F_{3/2, 2} & -(F_{3/2, 2}+F_{5/2, 2}) & F_{5/2, 2} & & \vdots\\\
0 & \ddots & \ddots & \ddots & \\\
\vdots & & & & 0 \\\
0 & \cdots & 0 & F_{n-1/2, n} & -F_{n-1/2, n}
\end{pmatrix}\ .$$
Including an imposed zonal flow U_j(y) in each layer, the equations of motion are:
$$\partial_t q_j + \mathsf{J}(\psi_j, q_j ) + (U_j - \partial_y\psi_j) \partial_x Q_j + U_j \partial_x q_j + (\partial_y Q_j)(\partial_x \psi_j) = -\delta_{j, n} \mu \nabla^2 \psi_n - \nu (-1)^{n_\nu} \nabla^{2n_\nu} q_j,$$
with
$$\partial_y Q_j \equiv \beta - \partial_y^2 U_j - (1-\delta_{j,1}) F_{j-1/2, j} (U_{j-1} - U_j) - (1 - \delta_{j,n}) F_{j+1/2, j} (U_{j+1} - U_j) + \delta_{j,n} \partial_y \eta \ , \\\
\partial_x Q_j \equiv \delta_{j,n} \partial_x \eta \ .$$
The eddy kinetic energy in each layer and the eddy potential energy that corresponds to each
fluid interface is computed via energies():
GeophysicalFlows.MultilayerQG.energies
The lateral eddy fluxes in each layer and the vertical fluxes across fluid interfaces are
computed via fluxes():
GeophysicalFlows.MultilayerQG.fluxes
Matrices \mathbb{S}_{\boldsymbol{k}} as well as \mathbb{S}^{-1}_{\boldsymbol{k}} are included
in params as params.S and params.S⁻¹ respectively. Additionally, the background PV gradients
\partial_x Q and \partial_y Q are also included in the params as params.Qx and params.Qy.
You can get \widehat{\psi}_j from \widehat{q}_j with streamfunctionfrompv!(psih, qh, params, grid),
while to get \widehat{q}_j from \widehat{\psi}_j you need to call pvfromstreamfunction!(qh, psih, params, grid).
The equations of motion are time-stepped forward in Fourier space:
$$\partial_t \widehat{q}_j = - \widehat{\mathsf{J}(\psi_j, q_j)} - \widehat{U_j \partial_x Q_j} - \widehat{U_j \partial_x q_j}
+ \widehat{(\partial_y \psi_j) \partial_x Q_j} - \widehat{(\partial_x\psi_j)(\partial_y Q_j)} + \delta_{j,n} \mu k^{2} \widehat{\psi}_n - \nu k^{2n_\nu} \widehat{q}_j \ .$$
In doing so the Jacobian is computed in the conservative form: \mathsf{J}(f,g) = \partial_y [ (\partial_x f) g] -\partial_x[ (\partial_y f) g].
Equations are formulated using $q_j$ as the state variables, i.e., sol = qh.
Thus:
$$\begin{aligned}
\mathcal{L} & = - \nu k^{2n_\nu} \ , \\\
\mathcal{N}(\widehat{q}_j) & = - \widehat{\mathsf{J}(\psi_j, q_j)} - \widehat{U_j \partial_x Q_j} - \widehat{U_j \partial_x q_j}
+ \widehat{(\partial_y \psi_j)(\partial_x Q_j)} - \widehat{(\partial_x \psi_j)(\partial_y Q_j)} + \delta_{j,n} \mu k^{2} \widehat{\psi}_n\ .
\end{aligned}$$
examples/multilayerqg_2layer.jl: A script that simulates the growth and equilibration of baroclinic eddy turbulence in the Phillips 2-layer model.