Merge pull request #114 from m2lines/textual_changes

updated the first few notebooks on gcm analogues and parameterization.

notebooks/L96-two-scale-description.ipynb

@@ -25,7 +25,10 @@
     "&= - cbY_{j+1,k} \\left( Y_{j+2,k} - Y_{j-1,k} \\right) - c Y_{j,k} + \\frac{hc}{b} X_k\n",
     "\\end{align}\n",
     "\n",
-    "where $X_k$, $k=1,\\ldots,K$, denotes $K$ slow (or low-frequency) variables and $Y_{j,k}$ , $j=1,\\ldots,J$ denotes $J*K$ fast (or high-frequency) variables. Each slow $X_k$ is coupled with $J$ fast $Y_{j,k}$ variables. The slow equations are coupled to the fast equations via a coupling term, $\\sum_{j=0}^{J-1} Y_{j,k}$, which sums over the $J$ fast variables corresponding to a particular $k$. On the other hand, each fast equation is forced by a coupling term, $\\frac{hc}{b} X_k$, that depends on the slow variable corresponding that particular $k$. The coupling term and the fast time-scale equation depend on three key parameters: $b$, $c$ and $h$. Here $b$ determines the magnitude of the nonlinear interactions among the fast variables, $c$ controls how rapidly the fast variables fluctuate compared to the slow variables and, $h$ governs the strength of the coupling between the slow and fast variables. Moreover, the slow time-scale equation is forced by the parameter $F$, whose value determines the chaotic behaviour of the system. E.g. {cite:p}`Wilks2005`.\n",
+    "where $X_k$, $k=1,\\ldots,K$, denotes $K$ slow (or low-frequency) variables, and $Y_{j,k}$ , $j=1,\\ldots,J$ denotes $J*K$ fast (or high-frequency) variables. \n",
+    "The slow equations are coupled to the fast equations via a coupling term, $\\sum_{j=0}^{J-1} Y_{j,k}$, which sums over the $J$ fast variables corresponding to a particular $k$. On the other hand, each fast equation is forced by a coupling term, $\\frac{hc}{b} X_k$, that depends on the slow variable corresponding to that particular $k$. \n",
+    "\n",
+    "The evolution of this two-time scale system depend on three key parameters: $b$, $c$ and $h$. Here $b$ determines the magnitude of the nonlinear interactions among the fast variables, $c$ controls how rapidly the fast variables fluctuate compared to the slow variables and, $h$ governs the strength of the coupling between the slow and fast variables. Moreover, the slow time-scale equation is forced by the parameter $F$, whose value determines the chaotic behaviour of the system. E.g. {cite:p}`Wilks2005`.\n",
     "\n",
     "The chaotic dynamical system L96 is very useful for testing different numerical methods in atmospheric modeling thanks to its transparency, low computational cost and simplicity compared to Global Climate Models (GCM). The interaction between variables of different scales makes the L96 model of particular interest when evaluating new parameterization methodologies. As such, it was used in assessing different techniques that were later incorporated into GCMs ({cite}`Crommelin2008`, {cite}`Dorrestijn2013`).\n",
     "\n",
@@ -290,7 +293,14 @@
     "# (Re-running this cell will give different answers after the first run)\n",
     "X2, Y2, t = M.run(0.005, 5, store=True)\n",
     "\n",
-    "print(\"Mean absolute difference =\", np.abs(X2 - X).mean() + np.abs(Y2 - Y).mean())"
+    "print(\n",
+    "    \"Mean absolute difference 1st run=\", np.abs(X2 - X).mean() + np.abs(Y2 - Y).mean()\n",
+    ")\n",
+    "\n",
+    "X2, Y2, t = M.run(0.005, 5, store=True)\n",
+    "print(\n",
+    "    \"Mean absolute difference 2nd run=\", np.abs(X2 - X).mean() + np.abs(Y2 - Y).mean()\n",
+    ")"
    ]
   },
   {
@@ -310,9 +320,7 @@
     "    L96(36, 10, F=10, dt=0.005)\n",
     "    .set_state(s(M.k, M.K) * (s(M.k, M.K) - 1) * (s(M.k, M.K) + 1), 0 * M.j)\n",
     "    .run(0.005, 5)\n",
-    ")\n",
-    "\n",
-    "print(\"Mean absolute difference =\", np.abs(X3 - X).mean() + np.abs(Y3 - Y).mean())"
+    ")"
    ]
   }
  ],
@@ -332,7 +340,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.16"
+   "version": "3.10.9"
   }
  },
  "nbformat": 4,

notebooks/estimating-gcm-parameters.ipynb

@@ -569,7 +569,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.16"
+   "version": "3.10.9"
   }
  },
  "nbformat": 4,