adds the long-awaited doc for BarotropicQGQL

docs/src/modules/barotropicqgql.md

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+# BarotropicQGQL Module
+
+### Basic Equations
+
+This module solves the *quasi-linear* quasi-geostrophic barotropic vorticity equation on a beta-plane of variable fluid depth $H-h(x,y)$. 
+Quasi-linear refers to the dynamics that *neglect* the eddy--eddy interactions in the eddy evolution equation after an eddy--mean flow decomposition, e.g., 
+
+$$\phi(x, y, t) = \overline{\phi}(y, t) + \phi'(x,y,t) ,$$
+
+where overline above denotes a zonal mean, $\overline{\phi}(y, t) = \int \phi(x, y, t)\,\mathrm{d}x/L_x$ and prime denotes deviations from the zonal mean. This approximation is used in many process-model studies of zonation, e.g., 
+
+- Farrell, B. F. and Ioannou, P. J. (2003). Structural stability of turbulent jets. *J. Atmos. Sci.*, **60**, 2101-2118.
+- Tobias, S. M. and Marston, J. B. (2013). Direct statistical simulation of out-of-equilibrium jets. *Phys. Rev. Lett.*, **110 (10)**, 104502.
+- Constantinou, N. C. (2014). Emergence and equilibration of jets in beta-plane turbulence: applications of Stochastic Structural Stability Theory. *J. Atmos. Sci.*, **71 (5)**, 1818-1842.
+- Parker, J. B. and Krommes, J. A. (2013). Zonal flow as pattern formation. *Phys. Plasmas*, **20**, 100703.
+
+
+As in the [BarotropicQG module](barotropicqg.md), the flow is obtained through a streamfunction $\psi$ as $(u, \upsilon) = (-\partial_y\psi, \partial_x\psi)$. All flow fields can be obtained from the quasi-geostrophic potential vorticity (QGPV). Here the QGPV is
+
+$$\underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{(\partial_x \upsilon
+	- \partial_y u)}_{\text{relative vorticity}} +
+	\underbrace{\frac{f_0 h}{H}}_{\text{topographic PV}}.$$
+
+The dynamical variable is the component of the vorticity of the flow normal to the plane of motion, $\zeta\equiv \partial_x \upsilon- \partial_y u = \nabla^2\psi$. Also, we denote the topographic PV with $\eta\equiv f_0 h/H$. After we apply the eddy-mean flow decomposition above, the QGPV dynamics are:
+
+$$\partial_t \overline{\zeta} + \mathsf{J}(\overline{\psi}, \underbrace{\overline{\zeta} + \overline{\eta}}_{\equiv \overline{q}}) + \overline{\mathsf{J}(\psi', \underbrace{\zeta' + \eta'}_{\equiv q'})} = \underbrace{-\left[\mu + \nu(-1)^{n_\nu} \nabla^{2n_\nu}
+\right] \overline{\zeta} }_{\textrm{dissipation}} \ .$$
+
+$$\partial_t \zeta' + \mathsf{J}(\psi', \overline{q}) + \mathsf{J}(\overline{\psi}, q') + \underbrace{\mathsf{J}(\psi', q') - \overline{\mathsf{J}(\psi', q')}}_{\textrm{EENL}} + 
+\beta\partial_x\psi' = \underbrace{-\left[\mu + \nu(-1)^{n_\nu} \nabla^{2n_\nu}
+\right] \zeta'}_{\textrm{dissipation}} + f\ .$$
+
+where $\mathsf{J}(a, b) = (\partial_x a)(\partial_y b)-(\partial_y a)(\partial_x b)$. On the right hand side, $f(x,y,t)$ is forcing (which is assumed to have zero mean, $\overline{f}=0$), $\mu$ is linear drag, and $\nu$ is hyperviscosity. Plain old viscosity corresponds to $n_{\nu}=1$. The sum of relative vorticity and topographic PV is denoted with $q\equiv\zeta+\eta$.
+
+*Quasi-linear* dynamics **neglect the term eddy-eddy nonlinearity (EENL) term** above.
+
+### Implementation
+
+The equation is time-stepped forward in Fourier space:
+
+$$\partial_t \widehat{\zeta} = - \widehat{\mathsf{J}(\psi, q)}^{\textrm{QL}} +\beta\frac{\mathrm{i}k_x}{k^2}\widehat{\zeta} -\left(\mu
++\nu k^{2n_\nu}\right) \widehat{\zeta}  + \widehat{f}\ .$$
+
+In doing so the Jacobian is computed in the conservative form: $\mathsf{J}(f,g) =
+\partial_y [ (\partial_x f) g] -\partial_x[ (\partial_y f) g]$. The superscript QL in the Jacobian term above denotes that remove triad interactions that correspond to the EENL term.
+
+Thus:
+
+$$\mathcal{L} = \beta\frac{\mathrm{i}k_x}{k^2} - \mu - \nu k^{2n_\nu}\ ,$$
+$$\mathcal{N}(\widehat{\zeta}) = - \mathrm{i}k_x \mathrm{FFT}(u q)^{\textrm{QL}}-
+	\mathrm{i}k_y \mathrm{FFT}(\upsilon q)^{\textrm{QL}}\ .$$
+
+
+## Examples
+
+- `examples/barotropicqgql_betaforced.jl`: A script that simulates forced-dissipative quasi-linear quasi-geostrophic flow on a beta-plane demonstrating zonation. The forcing is temporally delta-correlated and its spatial structure is isotropic with power in a narrow annulus of total radius `kf` in wavenumber space. This example demonstrates that the anisotropic inverse energy cascade is not necessary for zonation.