Prof. Dr. Markus Meier
\n",
" Leibniz Institute for Baltic Sea Research Warnemünde (IOW)
\n",
" E-Mail: markus.meier@io-warnemuende.de
Figure 1: Probability density functions for different potentials of stochastics processes. The process are described by the, from left to right: linear damping model, asymmetric feedback model, non-linear damping model, non-linear model.
\n", "```\n", "\n", "#### 1. Linear damping model\n", "- pdf is a normal distribution\n", "- potential and pdf shown in left column of Fig. 1\n", "```{math}\n", "\\frac{dX}{dt} = -c*X+f,~~~~~c>0\n", "```\n", "- example: slab ocean model for the sea surface temperature (SST). change in SST for a constant height $h$ depends on the atmospheric forcing:\n", "```{math}\n", ":label: eq4\n", "\\frac{dSST}{dt} = \\frac{1}{c_ph}F_{atmos}, ~~~~~ c_p=4*10^6\\frac{J}{Km^3},\n", "```\n", "- while the atmospheric forcing is linearly dependent of the SST itself:\n", "```{math}\n", ":label: eq5\n", "F_{atmos}=c_A(T_{atmos}-SST), ~~~~~ c_A=40\\frac{W}{Km^2}\n", "```\n", "- inserting [](#eq5) into [](#eq4) yields:\n", "```{math}\n", "\\frac{dSST}{dt} = -cSST+cT_{atmos},~~~~~\\mathrm{with}~c=\\frac{c_A}{c_ph}\n", "```\n", "- the time evolution of the SST is in the SST itself if the atmospheric temperature $T_{atmos}$ is white noise. this is called an AR(1)-process (more about AR-processes later in this lecture)\n", "\n", "#### 2. Asymmetric feedback model\n", "- pdf is skewed\n", "- potential and pdf shown in second to left column of Fig. 1\n", "```{math}\n", "\\frac{dX}{dt} = c_0 - c_1X-c_2X^2+f,~~~~~c_i>0\n", "```\n", "- if the damping is stronger for deviations to the larger side of the equilibrium state than for deviations to the lower side, the pdf will be positively skewed and deviations to smaller values are more likey than deviations to larger values\n", "- Fig. 2 shows the pdf of some exemplary advection variables\n", "```{figure} figures/L14/L14_2_advection.PNG\n", "---\n", "width: 40%\n", "---\n", "Figure 2: Some pdfs of observed daily mean climate variables.
\n", "```\n", "- Fig. 3 shows the pdf of NINO3 temperatures\n", "```{figure} figures/L14/L14_3_elnino.PNG\n", "---\n", "width: 25%\n", "---\n", "Figure 3: Pdf of El Nino monthly mean SST.
\n", "```\n", "\n", "#### 3. Non-linear damping model\n", "- pdf with kurtosis\n", "- potential and pdf shown in second to right column of Fig. 1\n", "```{math}\n", "\\frac{dX}{dt} = -cX^3+f,~~~~~c>0\n", "```\n", "- non-linear but symmetrical damping, negative kurtosis causes a pdf which is less peaked than the Gaussian distribution\n", "\n", "#### 4. Non-linear model\n", "- pdf is bimodal\n", "- potential and pdf shown in right column of Fig. 1\n", "```{math}\n", "\\frac{dX}{dt} =c_1X-c_3X^3+f,~~~~~c_i>0\n", "```\n", "- two equilibria caused by deterministic cycles (daily, annual), pdf of a sine function is bimodal\n", "\n", "\n", "## Autoregressive (Markov) processes\n", "- dynamics of many physical processes can be approximated by first-, second-, or sometimes higher-order ordinary linear differential equations:\n", "```{math}\n", ":label: ar\n", "a_2\\frac{d^2x(t)}{dt^2} + a_1 \\frac{dx(t)}{dt} + a_0x(t) = z(t)\n", "```\n", "- with $z(t)$ deribing the external forcing. if the external forcing $z(t)$ is a white noise process, then equation [](#ar) defines an auto-regressive process of the second order, short AR(2)-process\n", "- time discretization for numerical implementation:\n", "```{math}\n", "x_t=\\alpha_1x_{t-1}+\\alpha_2x_{t-2}+\\xi_t\n", "```\n", "- with the prefactors $\\alpha_1=\\frac{a_1+2a_2}{a_0+a_1+a_2}$, $\\alpha_2=\\frac{-a_2}{a_0+a_1+a_2}$ and $\\xi_t=\\frac{1}{a_0+a_1+a_2}z_t$\n", "- in general: $\\mathbf{X_t}:t \\in \\mathbb{Z}$ is an auto-regressive process of order p if there exists real constants $\\alpha_k~(k=0,1,...,p)$ with $\\alpha_k$ not equal to 0 and a white noise process $\\mathbf{Z_t}:t\\in\\mathbb{Z}$ such that:\n", "```{math}\n", "\\mathbf{X_t} = \\alpha_0 + \\sum_{k=1}^p\\alpha_k\\mathbf{X_{t-k}}+\\mathbf{Z_t}\n", "```\n", "- analytically an AR(p)-process is described by:\n", "```{math}\n", "a_0x(t) + \\sum_{k=1}^p a_k\\frac{d}{dt^k}x(t) = z(t)\n", "```\n", "- we assume $\\mu=0 \\Rightarrow \\alpha_0=0$. variance of AR(p)-processes is given by:\n", "```{math}\n", "Var(\\mathbf{X_t}) = \\frac{Var(\\mathbf{Z_t})}{1- \\sum_{k=1}^p\\alpha_k\\rho_k},\n", "```\n", "- with $\\rho_k$ the auto-correlation function of $\\mathbf{X_t}$:\n", "```{math}\n", ":label: rho\n", "\\rho_k = \\frac{Covar(\\mathbf{X_{t-k}X_t})}{Var(\\mathbf{X_t})}\n", "```\n", " \n", "#### AR(1)-processes\n", "- AR(1)-processes can be described by:\n", "```{math}\n", "\\mathbf{X_t} = \\alpha_1 \\mathbf{X_{t-1}} + \\mathbf{z_t},~~~~~\\alpha_1 \\in ]0,1[\n", "```\n", "- with a variance of\n", "```{math}\n", "Var(\\mathbf{X_t}) = \\frac{\\sigma_z^2}{1-\\alpha_1^2}\n", "```\n", "- $\\rho_1 = \\alpha_1$ lag(1) auto-correlation\n", "```{math}\n", "\\frac{dx(t)}{dt} = a_1x(t)+z(t),\n", "```\n", "- with the damping parameter\n", "```{math}\n", "a_1 = \\frac{\\alpha_1-1}{\\alpha_1}\n", "```\n", "- noise generates low-frequency variability\n", "```{figure} figures/L14/L14_4_AR1.PNG\n", "---\n", "width: 40%\n", "---\n", "Figure 4: Different realization of AR(1)-processes with different $\\alpha_1=0.5$, $0.9$, $0.99$ but with identical unit variance normal white noise processes $\\mathbf{Z_t}$. Note the different y-axis scaling.
\n", "```\n", "\n", "#### AR(2)-processes\n", "- AR(1)-processes can be described by:\n", "```{math}\n", "\\mathbf{X_t} = \\alpha_1 \\mathbf{X_{t-1}} + \\alpha_2 \\mathbf{X_{t-2}} + \\mathbf{z_t}\n", "```\n", "- stationary:\n", "```{math}\n", "\\begin{align*}\n", "\\alpha_2 + \\alpha_1 &< 1\\\\\n", "\\alpha_2 - \\alpha_1 &< 1\\\\\n", "|\\alpha_2| &< 1\\\\\n", "\\end{align*}\n", "```\n", "- AR(2)-process:\n", "```{math}\n", "\\alpha_2 \\neq \\alpha_1^2\n", "```\n", "- $\\alpha_2$ can only show oscillatory behavior if $\\alpha_2<0$\n", "```{figure} figures/L14/L14_5_AR2.PNG\n", "---\n", "width: 40%\n", "---\n", "Figure 5: Different realization of AR(2)-processes with different $\\alpha_1$ and $\\alpha_2$, but with identical unit variance normal white noise processes $\\mathbf{Z_t}$. Note the different y-axis scaling.
\n", "```\n", "- damped oscillating system, which can, unlike the AR(1)-process, have a preferred time scale at which the system oscillates if driven white noise, e.g. El Nino Southern Oscillation" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.6" } }, "nbformat": 4, "nbformat_minor": 5 }