Merge 3e43aac into 204e04f

docs/make.jl

@@ -20,6 +20,7 @@ sitename = "GeophysicalFlows.jl",
             "Modules" => Any[
               "modules/twodturb.md",
               "modules/barotropicqg.md",
+              "modules/barotropicqgql.md",
               "modules/multilayerqg.md"
             ],
             "DocStrings" => Any[

docs/src/modules/barotropicqg.md

@@ -1,9 +1,5 @@
 # BarotropicQG Module
 
-```math
-\newcommand{\J}{\mathsf{J}}
-```
-
 ### Basic Equations
 
 This module solves the quasi-geostrophic barotropic vorticity equation on a beta-plane of variable fluid depth $H-h(x,y)$. 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
@@ -14,20 +10,20 @@ $$\underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{(\partial_x \up
 
 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 $\eta\equiv f_0 h/H$. Thus, the equation solved by the module is:
 
-$$\partial_t \zeta + \J(\psi, \underbrace{\zeta + \eta}_{\equiv q}) +
+$$\partial_t \zeta + \mathsf{J}(\psi, \underbrace{\zeta + \eta}_{\equiv q}) +
 \beta\partial_x\psi = \underbrace{-\left[\mu + \nu(-1)^{n_\nu} \nabla^{2n_\nu}
 \right] \zeta }_{\textrm{dissipation}} + f\ .$$
 
-where $\J(a, b) = (\partial_x a)(\partial_y b)-(\partial_y a)(\partial_x b)$. On the right hand side, $f(x,y,t)$ is forcing, $\mu$ 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 $q\equiv\zeta+\eta$.
+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, $\mu$ 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 $q\equiv\zeta+\eta$.
 
 ### Implementation
 
 The equation is time-stepped forward in Fourier space:
 
-$$\partial_t \widehat{\zeta} = - \widehat{\J(\psi, q)} +\beta\frac{\mathrm{i}k_x}{k^2}\widehat{\zeta} -\left(\mu
+$$\partial_t \widehat{\zeta} = - \widehat{\mathsf{J}(\psi, q)} +\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: $\J(f,g) =
+In doing so the Jacobian is computed in the conservative form: $\mathsf{J}(f,g) =
 \partial_y [ (\partial_x f) g] -\partial_x[ (\partial_y f) g]$.
 
 Thus:
@@ -39,10 +35,10 @@ $$\mathcal{N}(\widehat{\zeta}) = - \mathrm{i}k_x \mathrm{FFT}(u q)-
 
 ## Examples
 
-- `examples/barotropicqg/decayingbetaturb.jl`: An script that simulates decaying quasi-geostrophic flow on a beta-plane demonstrating zonation.
+- `examples/barotropicqg_betadecay.jl`: A script that simulates decaying quasi-geostrophic flow on a beta-plane demonstrating zonation.
 
-- `examples/barotropicqg/forcedbetaturb.jl`: An script that simulates forced-dissipative quasi-geostrophic flow on a beta-plane demonstrating zonation. The forcing is temporally delta-corraleted and its spatial structure is isotropic with power in a narrow annulus of total radius `kf` in wavenumber space.
+- `examples/barotropicqg_betaforced.jl`: A script that simulates forced-dissipative 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.
 
-- `examples/barotropicqg/ACConelayer.jl`: A script that simulates barotropic quasi-geostrophic flow above topography reproducing the results of the paper by
+- `examples/barotropicqg_acc.jl`: A script that simulates barotropic quasi-geostrophic flow above topography reproducing the results of the paper by
 
   > Constantinou, N. C. (2018). A barotropic model of eddy saturation. *J. Phys. Oceanogr.*, **48 (2)**, 397-411.

docs/src/modules/barotropicqgql.md

@@ -0,0 +1,56 @@
+# 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.

docs/src/modules/multilayerqg.md

@@ -1,9 +1,5 @@
 # MultilayerQG Module
 
-```math
-\newcommand{\J}{\mathsf{J}}
-```
-
 ### 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 $(u_j, \upsilon_j) = (-\partial_y\psi_j, \partial_x\psi_j)$, $j=1,...,n$, where $n$ is the number of fluid layers.
@@ -52,7 +48,7 @@ where
 Including an imposed zonal flow $U_j(y)$ in each layer the equations of motion are:
 
 ```math
-\partial_t q_j + \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,
+\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
@@ -100,16 +96,16 @@ You can get $\widehat{\psi}_j$ from $\widehat{q}_j$ with `streamfunctionfrompv!(
 The equations are time-stepped forward in Fourier space:
 
 ```math
-\partial_t \widehat{q}_j = - \widehat{\J(\psi_j, q_j)}  - \widehat{U_j \partial_x Q_j} - \widehat{U_j \partial_x q_j}
+\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: $\J(f,g) =
+In doing so the Jacobian is computed in the conservative form: $\mathsf{J}(f,g) =
 \partial_y [ (\partial_x f) g] -\partial_x[ (\partial_y f) g]$.
 
 
 Thus:
 
 $$\mathcal{L} = - \nu k^{2n_\nu}\ ,$$
-$$\mathcal{N}(\widehat{q}_j) = - \widehat{\J(\psi_j, q_j)} - \widehat{U_j \partial_x Q_j} - \widehat{U_j \partial_x q_j}
+$$\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\ .$$