multilayerqg.md

MultilayerQG Module

Basic Equations

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

$${\mathrm{QGPV}}_j=q_j+\underset{\text{planetary PV}}{\underbrace{f_0+\beta y}}+{\delta }_{j,n}\underset{\text{topographic PV}}{\underbrace{\frac{f_{0}h}{H_n}}},j=1,\dots ,n.$$

where q_j incorporates the relative vorticity in each layer \nabla^2\psi_j and the vortex stretching terms:

$$q_1={\mathrm{\nabla }}^2{\psi }_1+F_{3/2,1}({\psi }_2-{\psi }_1),q_j={\mathrm{\nabla }}^2{\psi }_j+F_{j-1/2,j}({\psi }_{j-1}-{\psi }_j)+F_{j+1/2,j}({\psi }_{j+1}-{\psi }_j),j=2,\dots ,n-1,q_n={\mathrm{\nabla }}^2{\psi }_n+F_{n-1/2,n}({\psi }_{n-1}-{\psi }_n).$$

with

$$F_{j+1/2,k}=\frac{f_0^2}{g_{j+1/2}^{\prime }{H}_k}\text{and}{g}_{j+1/2}^{\prime }=g\frac{{\rho }_{j+1}-{\rho }_j}{{\rho }_{j+1}}.$$

In view of the relationships above, when we convert to Fourier space q's and \psi's are related via the matrix equation

$$\left(\begin{array}{c}{\hat{q}}_{\mathit{k},1}\\ ⋮\\ {\hat{q}}_{\mathit{k},n}\end{array}\right)=\underset{\equiv {\mathbb{S}}_{\mathit{k}}}{\underbrace{(-|\mathit{k}{|}^2\mathbb{1}+\mathbb{F})}}\left(\begin{array}{c}{\hat{\psi }}_{\mathit{k},1}\\ ⋮\\ {\hat{\psi }}_{\mathit{k},n}\end{array}\right)$$

where

$$\mathbb{F}\equiv \left(\begin{array}{ccccc}-F_{3/2,1}& F_{3/2,1}& 0& \cdots & 0\\ F_{3/2,2}& -(F_{3/2,2}+F_{5/2,2})& F_{5/2,2}& & ⋮\\ 0& \ddots & \ddots & \ddots & \\ ⋮& & & & 0\\ 0& \cdots & 0& F_{n-1/2,n}& -F_{n-1/2,n}\end{array}\right).$$

Including an imposed zonal flow U_j(y) in each layer, the equations of motion are:

$${\partial }_t{q}_j+\mathsf{J}({\psi }_j,q_j)+(U_j-{\partial }_y{\psi }_j){\partial }_x{Q}_j+U_j{\partial }_x{q}_j+({\partial }_y{Q}_j)({\partial }_x{\psi }_j)=-{\delta }_{j,n}\mu {\mathrm{\nabla }}^2{\psi }_n-\nu (-1{)}^{n_{\nu }}{\mathrm{\nabla }}^{2n_{\nu }}{q}_j,$$

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

Implementation

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:

$${\partial }_t{\hat{q}}_j=-\widehat{\mathsf{J}({\psi }_j,q_j)}-\widehat{U_j{\partial }_x{Q}_j}-\widehat{U_j{\partial }_x{q}_j}+\widehat{({\partial }_y{\psi }_j){\partial }_x{Q}_j}-\widehat{({\partial }_x{\psi }_j)({\partial }_y{Q}_j)}+{\delta }_{j,n}\mu k^2{\hat{\psi }}_n-\nu k^{2n_{\nu }}{\hat{q}}_j.$$

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:

$$\begin{array}{rl}\mathcal{L}& =-\nu k^{2n_{\nu }},\\ \mathcal{N}({\hat{q}}_j)& =-\widehat{\mathsf{J}({\psi }_j,q_j)}-\widehat{U_j{\partial }_x{Q}_j}-\widehat{U_j{\partial }_x{q}_j}+\widehat{({\partial }_y{\psi }_j)({\partial }_x{Q}_j)}-\widehat{({\partial }_x{\psi }_j)({\partial }_y{Q}_j)}+{\delta }_{j,n}\mu k^2{\hat{\psi }}_n.\end{array}$$

Examples

• examples/multilayerqg_2layer.jl: A script that simulates the growth and equilibration of baroclinic eddy turbulence in the Phillips 2-layer model.