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The wind-driven ocean circulation: Applying dynamical systems theory to a climate problem

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  • The large-scale, near-surface flow of the mid-latitude oceans is dominated by the presence of a larger, anticyclonic and a smaller, cyclonic gyre. The two gyres share the eastward extension of western boundary currents, such as the Gulf Stream or Kuroshio, and are induced by the shear in the winds that cross the respective ocean basins. This physical phenomenology is described mathematically by a hierarchy of systems of nonlinear partial differential equations (PDEs). We study the low-frequency variability of this wind-driven, double-gyre circulation in mid-latitude ocean basins, subject to time-constant, purely periodic and more general forms of time-dependent wind stress. Both analytical and numerical methods of dynamical systems theory are applied to the PDE systems of interest. Recent work has focused on the application of non-autonomous and random forcing to double-gyre models. We discuss the associated pullback and random attractors and the non-uniqueness of the invariant measures that are obtained. The presentation moves from observations of the geophysical phenomena to modeling them and on to a proper mathematical understanding of the models thus obtained. Connections are made with the highly topical issues of climate change and climate sensitivity.

    Mathematics Subject Classification: Primary:37B55, 37H99, 65P30, 76U05; Secondary:34C26, 37G35, 37L30, 37L40, 65P20.


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  • Figure 1.  A map of the main oceanic currents: warm currents in red and cold ones in blue. Reproduced from [79], with permission from Elsevier

    Figure 2.  A satellite image of the sea surface temperature (SST) field over the northwestern North Atlantic (in false color, from the U.S. National Oceanic and Atmospheric Administration), together with a sketch of the associated double-gyre circulation (white arrows). A highly simplified and smoothed view of the amount of potential vorticity injected into the ocean circulation by the equatorial trade winds, the mid-latitudes' prevailing westerlies and the polar easterlies is shown in the sketch to the right. Reproduced from [79], with permission from Elsevier

    Figure 3.  Generic bifurcation diagram for the barotropic QG model of the double-gyre problem: the asymmetry of the solution is plotted versus the intensity of the wind stress $\tau$. The streamfunction field is plotted for a steady-state solution associated with each of the three branches; positive values in red and negative ones in blue. After [170]

    Figure 4.  Pullback attractor (PBA) for a scalar linear equation, given by the solid black line. We wish to observe the PBA at times $t = t_1, t_2$, marked by dashed vertical lines, and consider the convergence to the straight line of orbits started at past times $s = s_1, s_2$; sample orbits for $s_1$ and $s_2$ are plotted in red and blue, respectively. Courtesy of M. D. Chekroun

    Figure 5.  Snapshot of the Lorenz [119] model's random attractor ${\mathcal A}(\omega)$ and of the corresponding sample measure $\mu_\omega$, for a given, fixed realization $\omega$. The figure corresponds to projection onto the $(y,z)$ plane, i.e. $\int \mu_{\omega} (x,y,z)\mbox{d}x$. One billion initial points have been used in both panels and the pullback attractor is computed for $t = 40$. The parameter values are the classical ones -- $r=28$, $s=10$, and $b=8/3$, while the time step is $\Delta t = 5\cdot 10^{-3}$. The color bar to the right of each panel is on a log-scale and quantifies the probability to end up in a particular region of phase space. Both panels use the same noise realization $\omega$ but with noise intensity (a) $\sigma=0.3$ and (b) $\sigma=0.5$. Notice the interlaced filamentary structures between highly (yellow) and moderately (red) populated regions; these structures are much more complex in panel (b), where the noise is stronger. Weakly populated regions cover an important part of the random attractor and are, in turn, entangled with (almost) zero-probability regions (black). After [36]

    Figure 6.  Four snapshots of the random attractor and sample measure supported on it, for the same parameter values as in Fig.5. The time interval $\Delta t$ between two successive snapshots -- moving from left to right and top to bottom -- is $\Delta t = 0.0875$. Note that the support of the sample measure may change quite abruptly, from time to time, cf. short video in [36,Supplementary Material] for details. Reproduced from [36], with permission from Elsevier

    Figure 7.  Ensemble behavior of forced solutions of the double-gyre ocean model of [150]. (a) Time dependence of the total forcing $1+\epsilon f(t)$, for $\epsilon = 0.2$. (b, c) Evolution of $N = 644$ initial states emanating from the subset $\Gamma$ in the $(\Psi_1, \Psi_3)$-plane for (b) $\gamma=0.96$ and (c) $\gamma=1.1$. (b', c') Corresponding time series of $P_{\Psi_3}$. Reproduced from [150], with the permission of the American Meteorological Society

    Figure 8.  Mean normalized distance $\sigma(\Psi_1,\Psi_3)$ for 15 000 trajectories of the double-gyre ocean model starting in the initial set $\Gamma$: (a) $\gamma=0.96$, and (b) $\gamma=1.1$. Reproduced from [150], with the permission of the American Meteorological Society

    Figure 9.  Long-periodic orbits and slow manifold of the coupled ocean-atmosphere model of [198], in a three-dimensional (3-D) projection onto the leading modes $(\psi_{{\rm a},1}, \psi_{{\rm o},2}, T_{{\rm o},2})$ of the atmospheric and oceanic streamfunction fields and that of the oceanic temperature field. (a) Long-periodic orbits of the coupled model, for the friction parameter $d=1 \times 10^{-8}$ s$^{-1}$ and several values of $C_{\rm o}$ (see legend). (b) Long-periodic orbits and LFV-dominated ones, for the radiative-input parameter $C_{\rm o} = 300$ Wm$^{-2}$ and several values of $d$: $5 \times 10^{-9}$ s$^{-1}$ (red), $1 \times 10^{-8}$ s$^{-1}$ (green), $2 \times 10^{-8}$ s$^{-1}$ (dark blue), $3 \times10^{-8}$ s$^{-1}$ (magenta), and $8 \times 10^{-8}$ s$^{-1}$ (light blue). After [198]

    Figure 10.  Low-frequency variability (LFV) of the coupled ocean-atmosphere model. Time series of geopotential height difference between locations ($\pi/n, \pi/4)$) and ($\pi/n, 3 \pi/4)$) of the model's nondimensional domain, for different values of meridional temperature gradient $C_{\rm o}$ and coupling coefficient $d$; this height difference plays the role of a North Atlantic Oscillation index in the model. (a) Chaotic but smooth trajectories living on a hypothetical slow attractor; and (b) strongly fluctuating trajectories that are not lying close to such a slow attractor. Reproduced from [198], with permission from Elsevier

    Figure 11.  Climate sensitivity for (a) an equilibrium model; and (b, c) a nonequilibrium model. Given a jump in a parameter, such as CO$_2$ concentration, only the mean global temperature $\overline T$ changes in (a), while in (b) it is also the period, amplitude and phase of a purely periodic oscillation, such as the seasonal cycle or the intrinsic ENSO cycle. Finally, in panel (c), it is also the character of the oscillation, whether deterministic or stochastically perturbed, which may change. After [71]

    Figure 12.  Time-dependent invariant measures of the [66] model; snapshots shown at three times $t$: (a) $t_1 = 19.23$ yr, (b) $t_2 = 20$ yr and (c) $t_3=20.833$ yr. After [40], with the authors' permision

    Figure 13.  Random dynamical systems (RDS) viewed as a flow on the bundle $X \times \Omega$ = "dynamical space" $\times$ "probability space." For a given state $x$ and realization $\omega$, the RDS $\varphi$ is such that $\Theta(t)(x,\omega) = (\theta(t)\omega,\varphi(t,\omega)x)$ is a flow on the bundle. Reproduced from [79], with permission from Elsevier

    Figure 14.  Schematic diagram of a random attractor ${\mathcal A}(\omega)$, where $\omega \in \Omega$ is a fixed realization of the noise. To be attracting, for every set $B$ of $X$ in a family $\mathfrak{B}$ of such sets, one must have $\lim_{t\to +\infty} {\text{dist}}(B(\theta(-t)\omega), {\mathcal A}(\omega)) = 0$ with $B(\theta(-t)\omega):=\varphi(t,\theta(-t)\omega) B$; to be invariant, one must have $\varphi(t,\omega) {\mathcal A}(\omega) = {\mathcal A}(\theta(t)\omega)$. This definition depends strongly on $\mathfrak{B}$; see [45] for more details. Reproduced from [79], with permission from Elsevier

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