Dynamics of potential vorticity staircase evolution and step mergers in a reduced model of beta-plane turbulence

A two-field model of potential vorticity (PV) staircase structure and dynamics relevant to both beta-plane and drift-wave plasma turbulence is studied numerically and analytically. The model evolves averaged PV whose flux is both driven by and regulates a potential enstrophy field, $\varepsilon$. The model employs a closure using a mixing length model. Its link to bistability, vital to staircase generation, is analyzed and verified by integrating the equations numerically. Long-time staircase evolution consistently manifests a pattern of metastable quasiperiodic configurations, lasting for hundreds of time units, yet interspersed with abrupt ($\mathrm{\Delta }t\ll 1$) mergers of adjacent steps in the staircase. The mergers occur at the staircase lattice defects where the pattern has not completely relaxed to a strictly periodic solution that can be obtained analytically. Other types of stationary solutions are solitons and kinks in the PV gradient and $\varepsilon$-profiles. The waiting time between mergers increases strongly as the number of steps in the staircase decreases. This is because of an exponential decrease in interstep coupling strength with growing spacing. The long-time staircase dynamics is shown numerically be determined by local interaction with adjacent steps. Mergers reveal themselves through the explosive growth of the turbulent PV flux, which, however, abruptly drops to its global constant value once the merger is completed.

Figure 1

Enstrophy production-dissipation term $R$ in Eqs. (7) and (8) as a function of enstrophy $\varepsilon$, shown for a fixed mean vorticity gradient $Q_y=35$ and ${\varepsilon }_0=3.55$.

Figure 2

Left-hand side of Eq. (10), $F\left(\xi \right)$, schematically drawn for three different values of parameter ${\varepsilon }_0$.

Figure 3

Part of parameter space in variables ${\varepsilon }_0,Q_y$ where the SC solutions are possible.

Figure 4

Generation of a one-step profile out of an unstable ground state superposed by small perturbations (short-dash lines). Perturbations are low-amplitude, so short scales are not seen clearly in the initial profile but they do not affect the profile evolution significantly. The Dirichlet boundary conditions are applied for $Q$ and $\varepsilon$ at both boundaries. The other parameters are ${\varepsilon }_0=3.8, L^2=2.3\times {10}^3, D=1.5$; the boundary conditions can be read from the plots.

Figure 5

The same plots as in Fig. 4 but for $L^2=900$. The other parameters are the same, except the turbulence level is kept at different levels on the right and left boundary (6.0 and 23.0, respectively). Initially, it is a linear function superposed by a sinusoidal with a short scale ($\sim 0.03)$ and small amplitude ($\approx 0.03)$. Since parameter $L$ is relatively small, the short-scale initial perturbations do not grow (cf. Fig. 10 below). As the midplane symmetry is broken, the initially formed profiles propagate to the right boundary. After $t\lesssim 1$, the $Q$-profile relaxes to one step at the right boundary and a shear layer at the left. This configuration persists in time, as the boundary conditions are consistent with a part of periodic stationary solution (Sec. 5) that fits into the integration domain.

Figure 6

“Oscillator's” pseudopotential and its phase plane. The top panel shows the function $W\left(\psi ,b\right)$ from Eq. (15) for the case when two maxima that correspond to two zeros of function $R\left(\varepsilon \right)$ in Eq. (8) are at the same value of $W=E_{\max }$. The maxima are the same for a specific value of the constant flux $b$ in Eq. (6). For $E<E_{max}$ solutions are periodic.

Figure 7

Illustration of a Maxwell transition rule, in a form of the production-dissipation function, $R\left(y\right)$. In order to connect the two stable equilibria, shown in Fig. 1, with one heteroclinic orbit, shown in Fig. 6, the areas above and below the abscissa must be equal.

Figure 8

Merger of steps shown as a surface plot of the mean vorticity $Q\left(t,x\right)$ with the Dirichlet boundary conditions, $\mathrm{\Delta }Q=6.2$, between the right and left boundary, ${\varepsilon }_0=10.38, L^2=1.5\times {10}^5, D=1.7$. Initially, as many as 12 steps are formed, but very rapidly ($t\lesssim 0.1)$ they merge to 7. The seven-step configuration lasts with no numerically significant changes up to the moment $t\approx 2$ when two shear layers near the walls disappear and the edge steps merge into respective walls. The remaining SC structure persists up to $t\simeq 100$ when the two edge steps merge with their neighbors.

Figure 9

Numerical solution of Eqs. (6) and (7) in long-time asymptotic regime, shown with the solid line. The parameters are the same as in Fig. 4. Exact analytic solution represented by the two branches of $y\left(\psi \right)$ in Eq. (18), shown with red and green squares. The insignificant deviations at the top and the bottom of the curve are due to the limited accuracy of numerical integration at the end points in Eq. (18).

Figure 10

Typical profile of $Q_y$ where FWHM (full width at half maximum) characterizes the width of shear layers (“humps” on $Q_y$ or “pedestals” on the $Q$-profile) and steps (troughs) on the plot. Here $D=1.5, L^2={10}^5, {\varepsilon }_0=3.7915, Q\left(1\right)=6.2$, and $\varepsilon \left(0,1\right)=10.38$.

Figure 11

Time history and details of the SC merger sequence that led to the profile shown in Fig. 10. Panels (a) and (b) show the enstrophy $\varepsilon$ and $Q_y$, while panels (c) and (d) show the mean potential vorticity $Q$ and its flux given by Eq. (13). The early relaxation phase $t\lesssim 0.01$ is removed from the plots to emphasize the flux strong variation during the later SC mergers. Abrupt drops in the $Q$-flux excess at the merger sites after initially slow and then explosive growth are clearly seen in panel (d). It is also notable from, e.g., panel (a) that a merger still produces some small effect on the nearest seemingly unaffected steps.

Figure 12

$Q$-flux as a function of time immediately before and during the merger event near $t\approx 0.09$ at the middle of integration domain at $y=0.5$ shown in Fig. 11. For the clarity of representation and a functional fit, shown in Fig. 13, the flux is adjusted by subtracting a reference value $F_B\approx 13.96088$, which is close to a globally averaged flux.

Figure 13

Fit of the flux in its self-similar phase of evolution for $0.070<t<0.086$. The fit is given by $F=F_0+B/{\left(t_0-t\right)}^{\alpha }$ with $t_0\approx 0.0863, B\approx 0.000806, \alpha \approx 0.879$, subtracted background $F_B\approx 13.96088$, a residual flux constant $F_0\approx -0.0171$.