Spatiotemporal dynamics of the covariant Lyapunov vectors of chaotic convection

We explore the spatiotemporal dynamics of the spectrum of covariant Lyapunov vectors for chaotic Rayleigh-Bénard convection. We use the inverse participation ratio to quantify the amount of spatial localization of the covariant Lyapunov vectors. The covariant Lyapunov vectors are found to be spatially localized at times when the instantaneous covariant Lyapunov exponents are large. The spatial localization of the Lyapunov vectors often occurs near defect structures in the fluid flow field. There is an overall trend of decreasing spatial localization of the Lyapunov vectors with increasing index of the vector. The spatial localization of the covariant Lyapunov vectors with positive Lyapunov exponents decreases an order of magnitude faster with increasing vector index than all of the remaining vectors that we have computed. We find that a weighted covariant Lyapunov vector is useful for the visualization and interpretation of the significant connections between the Lyapunov vectors and the flow field patterns.

Figure 1

A schematic of the square planform convection domain used in our numerical simulations and our convention for the Cartesian coordinates $(x,y,z)$. The aspect ratio of the domain is $\mathrm{\Gamma }=L/d$ where $L$ is the side length and $d$ and is the layer depth. A shallow layer of fluid is contained between a hot surface (bottom, red) and a cold surface (top, blue) which are held at constant temperature. All material surfaces are no-slip and the lateral sidewalls are perfectly insulating. The direction of gravity $g$ is vertically downward.

Figure 2

An image of a typical flow field from a numerical simulation of chaotic convection. The temperature is shown at the midplane $T(x,y,z=1/2,t)$ where red is hot rising fluid and blue is cool falling fluid. The color bar defines the scale used in the plot. ($R=5000,\sigma =1,\mathrm{\Gamma }=16$).

Figure 3

Different projections of the leading order Lyapunov vector for the conditions of Fig. 2. Color contours indicate the value of the different perturbation fields at the horizontal midplane ($z=1/2$) where red is a large positive value and blue is a large negative value. The black solid lines indicate the location of the convection rolls for reference. (a) Perturbation temperature $\delta T^{\left(1\right)}$, (b) perturbation velocity in the $x$ direction $\delta u^{\left(1\right)}$, (c) perturbation velocity in the $y$ direction $\delta v^{\left(1\right)}$, and (d) perturbation velocity in the $z$ direction $\delta w^{\left(1\right)}$.

Figure 4

Comparison of the temperature perturbation field $|\delta T^{\left(i\right)}|$ and the magnitude of the Lyapunov vector $\left|\right|{\vec{\xi }}^{\left(i\right)}\left|\right|$ using Eq. (9) for the leading order Lyapunov vector ($i=1$), the Lyapunov vector with a vanishing Lyapunov exponent ($i=12$), and a rapidly decaying Lyapunov vector ($i=141$). Color contours indicate the value of the different fields at the horizontal midplane ($z=1/2$) where red is a large value and blue is small value. The black solid lines indicate the location of the convection rolls for reference. (a) $|\delta T^{\left(1\right)}|$, (b) $\left|\right|{\vec{\xi }}^{\left(1\right)}\left|\right|$, (c) $|\delta T^{\left(12\right)}|$, (d) $\left|\right|{\vec{\xi }}^{\left(12\right)}\left|\right|$, (e) $|\delta T^{\left(141\right)}|$, and (f) $\left|\right|{\vec{\xi }}^{\left(141\right)}\left|\right|$.

Figure 5

The spatial variation of the leading order Lyapunov vector for $R=5000,\sigma =1$, and $\mathrm{\Gamma }=16$ at two different times. Color contours indicate the value of the perturbation temperature field $\delta T^{\left(1\right)}$ at the horizontal midplane ($z=1/2$) where red is a large positive value and blue is a large negative value. The black solid lines indicate the location of the convection rolls for reference. (a) At this time the localization is large and $C_1=0.11$. This is reflected by the spatial distribution of the vector where its magnitude is mostly concentrated in one region near the center. (b) Twenty time units later the same Lyapunov vector is quite delocalized. At this time, $C_1$ is two orders of magnitude smaller at $C_1=0.001$.

Figure 6

(a) The time variation of the leading order instantaneous Lyapunov exponent ${\tilde{\lambda }}_1\left(t\right)$. (b) The time variation of the localization of the leading order Lyapunov vector $C_1\left(t\right)$. The vertical dashed line at $t=89.3$ corresponds to the time when both ${\tilde{\lambda }}_1\left(t\right)$ and $C_1\left(t\right)$ exhibit a local peak in value. The spatial distribution of the Lyapunov vector for this case is shown in panel (c). The vertical dashed line at $t=98.5$ corresponds to an instance when ${\tilde{\lambda }}_1\left(t\right)$ and $C_1\left(t\right)$ are at local a minimum. The Lyapunov vector at this time is shown in panel (d). Color contours indicate the value of the perturbation temperature $\delta T^{\left(1\right)}$ at the horizontal midplane ($z=1/2$) where red is a large positive value and blue is a large negative value. The black solid lines indicate the location of the convection rolls. (Rayleigh number $R=5000$, Prandtl number $\sigma =1$, and aspect ratio $\mathrm{\Gamma }=16$).

Figure 7

The spatiotemporal features of the 12th Lyapunov vector where (a) shows the time variation of the instantaneous Lyapunov exponent ${\tilde{\lambda }}_{12}$, (b) shows the localization $C_{12}\left(t\right)$, and (c) and (d) show images of the Lyapunov vectors at times $t=89$ and $t=93$, respectively, which are also indicated by the vertical dashed lines. Simulation parameters and conventions are the same as Fig. 6.

Figure 8

The spatiotemporal features of the $141\text{st}$ Lyapunov vector where (a) shows the time variation of the instantaneous Lyapunov exponent ${\tilde{\lambda }}_{141}$, (b) shows the localization $C_{141}\left(t\right)$, and (c) and (d) show the Lyapunov vector at times $t=89$ and $t=98.5$, respectively, which are also indicated by the vertical dashed lines. Simulation parameters and conventions are the same as in Fig. 6.

Figure 9

The equal-time cross correlation of the localization $C_i\left(t\right)$ and the instantaneous Lyapunov exponent ${\tilde{\lambda }}_i\left(t\right)$. The circles (blue) are for $i\le 11$ which represent Lyapunov vectors with positive Lyapunov exponents ${\lambda }_i>0$ and the squares (red) are for $i\ge 12$ which represent the Lyapunov vectors with ${\lambda }_i\le 0$. The localization and the instantaneous Lyapunov exponent are positively correlated. The overall trend is a decreasing amount of correlation as $i$ is increased. These results have been normalized by the value of the equal-time correlation computed for the leading order Lyapunov vector $(i=1)$ which had a value of 0.0024.

Figure 10

The variation of the localization $C_i$ with the instantaneous Lyapunov exponent ${\tilde{\lambda }}_i$. Circles (blue) are for the leading order Lyapunov vector, squares (red) are for the 12th covariant Lyapunov vector, and triangles (green) are for the $141\text{st}$ covariant Lyapunov vector.

Figure 11

The time average of the localization of the covariant Lyapunov vectors ${〈C_i〉}_t$ where $i$ is the index of the Lyapunov vector. The time average is taken over 100 time units. (a) Results for $R=5000$ where the circles (blue) represent the first 11 Lyapunov exponents which are all positive. The squares (red) represent the remaining Lyapunov vectors which have Lyapunov exponents that are near zero or negative. The dashed line is a linear curve fit through the circles with slope ${\alpha }_1=-1.6\times {10}^{-4}$. The dash-dotted line is a curve fit through the squares with slope ${\alpha }_2=-1.3\times {10}^{-5}$. (b) Results for $5000\le R\le 9000$. The overall trend is decreasing localization with increasing order of the Lyapunov vector. For all Rayleigh numbers, the localization of the Lyapunov vectors associated with positive exponents falls off approximately ten times faster than the localization of the remaining vectors with increasing order $i$.

Figure 12

Comparison of the spatial variation of the leading order Lyapunov vector and the weighted covariant Lyapunov vector. Color represents the magnitude of the Lyapunov vector where red is large and blue is small. The black contour lines indicate the boundaries of the convection rolls. Each row shows results at a different instant of time where the left side, panels (a), (c), (e), illustrate the leading order Lyapunov vector and the right side, panels (b), (d), (f), represent the weighted Lyapunov vectors.