update paper for generation artefacts

.github/workflows/draft-pdf.yml

@@ -1,3 +1,5 @@
+name: JOSS Paper Generation
+
 on: [push]
 
 jobs:
@@ -7,12 +9,14 @@ jobs:
     steps:
       - name: Checkout
         uses: actions/checkout@v3
+
       - name: Build draft PDF
         uses: openjournals/openjournals-draft-action@master
         with:
           journal: joss
           # This should be the path to the paper within your repo.
           paper-path: paper.md
+
       - name: Upload
         uses: actions/upload-artifact@v1
         with:

paper.md

@@ -78,7 +78,10 @@ $$
     \partial_t n = c_1 \left(\phi - n \right)
                      - \left[\phi, n \right]
                      - \kappa_n \partial_y \phi
-                     - \nu \nabla^{2N} n  \\
+                     - \nu \nabla^{2N} n
+$$
+
+$$
     \partial_t \Omega = c_1 \left( \phi - n \right)
                                     - \left[ \phi, \Omega \right]
                                     - \nu \nabla^{2N} \Omega 
@@ -102,36 +105,34 @@ However, the use of the Arakawa Scheme for the Poisson brackets does allow the p
 
 The reason why the Hasegawa-Wakatani Model has been the de-facto testing bed for new methods are its verifiable statistically stationary properties for the complex turbulent system, such as the turbulent particle flux $\Gamma^n$, primary sink $\Gamma^c$, energy E, enstrophy U.
 
-$$
-    \Gamma^n       = - \iint{ \mathrm{d}^2 x \space \left(n \partial_y \phi\right) } \\
-    \Gamma^c       = c_1   \iint{ \mathrm{d}^2 x \space \left(n - \phi\right)^2} \\
-    E              = \small \frac{1}{2} \normalsize \iint{ \mathrm{d}^2 x \space \left(n^2 - \left|\nabla_\bot \phi \right|^2 \right)} \\
-    U              = \small \frac{1}{2} \normalsize \iint{ \mathrm{d}^2 x \space \left(n-\nabla_\bot^2  \phi\right)^2} = \small \frac{1}{2} \normalsize \iint{ \mathrm{d}^2 x \space \left(n-\Omega\right)^2}
-$$
+$$ \Gamma^n       = - \iint{ \mathrm{d}^2 x \space \left(n \partial_y \phi\right) } $$
+$$ \Gamma^c       = c_1   \iint{ \mathrm{d}^2 x \space \left(n - \phi\right)^2} $$
+$$  E              = \small \frac{1}{2} \normalsize \iint{ \mathrm{d}^2 x \space \left(n^2 - \left|\nabla_\bot \phi \right|^2 \right)} $$
+$$  U              = \small \frac{1}{2} \normalsize \iint{ \mathrm{d}^2 x \space \left(n-\nabla_\bot^2  \phi\right)^2} = \small \frac{1}{2} \normalsize \iint{ \mathrm{d}^2 x \space \left(n-\Omega\right)^2} $$
 
 These can be complemented with spectral definitions of these, or the characteristic phase shift $\delta(k_y)$ between the density $n$ and potential $\phi$:
 
-$$
-  \int{\mathrm{d} k_y \space \Gamma^n \small (k_y)}  \normalsize \space = - \int{\mathrm{d} k_y \space \left( i k_y   n\small (k_y)  \normalsize \phi^* \small (k_y)\normalsize \right) } \\
-  \delta \small (k_y)  \normalsize \space = - \mathrm{Im}\left( \mathrm{log}\left( n^*\small (k_y)  \normalsize \space \phi\small (k_y)  \normalsize  \right) \right) \\
-  E^N  \small (k_y)  \normalsize \space = \small \frac{1}{2}\normalsize \big| n \small (k_y) \normalsize  \big|^2 \\
-  E^V  \small (k_y)  \normalsize \space = \small \frac{1}{2}\normalsize \big| k_y \phi \small (k_y) \normalsize  \big|^2 
-$$
+
+$$  \int{\mathrm{d} k_y \space \Gamma^n \small (k_y)}  \normalsize \space = - \int{\mathrm{d} k_y \space \left( i k_y   n\small (k_y)  \normalsize \phi^* \small (k_y)\normalsize \right) } $$
+$$  \delta \small (k_y)  \normalsize \space = - \mathrm{Im}\left( \mathrm{log}\left( n^*\small (k_y)  \normalsize \space \phi\small (k_y)  \normalsize  \right) \right) $$
+$$  E^N  \small (k_y)  \normalsize \space = \small \frac{1}{2}\normalsize \big| n \small (k_y) \normalsize  \big|^2 $$
+$$  E^V  \small (k_y)  \normalsize \space = \small \frac{1}{2}\normalsize \big| k_y \phi \small (k_y) \normalsize  \big|^2 $$
+
 
 Beside the spectral verification, time-integration can be verified through the expected in- and out-flows in the turbulent phase:
-$$
-    \partial_t E   = \Gamma^N - \Gamma ^c - \mathfrak{D}^E  \\
-    \partial_t U   = \Gamma^N - \mathfrak{D}^U  \\ 
-$$
+
+$$    \partial_t E   = \Gamma^N - \Gamma ^c - \mathfrak{D}^E $$
+$$    \partial_t U   = \Gamma^N - \mathfrak{D}^U  $$
+
 
 Where $\mathfrak{D}^E$ and $\mathfrak{D}^U$ are sinks derived from the diffusion terms of the HW equations, namely:
 
-$$
-    \mathfrak{D}^E = \quad \iint{ \mathrm{d}^2 x \space (n \mathfrak{D^n} - \phi \mathfrak{D}^\phi)} \\ 
-    \mathfrak{D}^U = -     \iint{ \mathrm{d}^2 x \space (n - \Omega)(\mathfrak{D}^n - \mathfrak{D}^\phi)} \\
-    with \quad \mathfrak{D}^n \small (x,y) \normalsize  = \nu \nabla^{2N} n \quad and \quad 
-    \mathfrak{D}^\phi \small (x,y) \normalsize \space = \nu \nabla^{2N} \phi  
-$$
+
+$$    \mathfrak{D}^E = \quad \iint{ \mathrm{d}^2 x \space (n \mathfrak{D^n} - \phi \mathfrak{D}^\phi)} $$
+$$    \mathfrak{D}^U = -     \iint{ \mathrm{d}^2 x \space (n - \Omega)(\mathfrak{D}^n - \mathfrak{D}^\phi)} $$
+$$    with \quad \mathfrak{D}^n \small (x,y) \normalsize  = \nu \nabla^{2N} n \quad and \quad 
+    \mathfrak{D}^\phi \small (x,y) \normalsize \space = \nu \nabla^{2N} \phi  $$
+
 
 Note that it is the common practice across all reference texts to calculate the integral $\int\cdot$ as the average over a unit square $\langle \cdot \rangle$ in order to get comparable values for all properties.