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A stochastic structural stability theory model of the drift wave–zonal flow system
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10.1063/1.3258666
/content/aip/journal/pop/16/11/10.1063/1.3258666
http://aip.metastore.ingenta.com/content/aip/journal/pop/16/11/10.1063/1.3258666
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Initial jet formation by the rapid adjustment process starting from a state of strong turbulence for the cases (a) (no instability) and (b) (strong instability). Shown are eddy kinetic energy (dashed line) and mean zonal kinetic energy (solid line) as a function of time. The eddy field is limited to poloidal wavenumber and there is no stochastic excitation.

Image of FIG. 2.
FIG. 2.

Transient development of an equilibrium zonal jet. (a) Time development of the mean kinetic energy of the zonal flow, (solid line), the mean eddy kinetic energy (dashed line), and the total particle flux over the channel (dash-dotted line). (b) Zonal velocity as a function of the radial direction and time. (c) Eddy kinetic energy, , as a function of the radial direction and time. (d) Eddy induced zonal flow acceleration, , as a function of the radial direction and time. The parameters are , and the stochastic excitation has equivalent rms velocity of .

Image of FIG. 3.
FIG. 3.

Evolution of the zonal flow and its associated spectrum for the example in Fig. 2. Left panels: zonal flow structure at and at equilibrium. Center panels: phase speed and growth rate associated with the temporal eigenvalues of for the flow in the corresponding panel for poloidal wavenumber . The continuous line indicates the velocity interval spanned by the zonal flow. The flow is unstable when there are eigenvalues with . At equilibrium the instabilities have been stabilized. Right panels: the largest growth rate for a given poloidal wavenumber . At equilibrium the least stable mode corresponds to the gravest poloidal wavenumber.

Image of FIG. 4.
FIG. 4.

(a) Maximum zonal flow velocity as a function of stochastic excitation for . Stochastic excitation is measured by the that would have been maintained in the absence of a zonal flow. For the chosen parameters the critical forcing required to form zonal flows is . (b) Corresponding equilibrium zonal zonal flows: the larger velocity corresponds to forcing denoted with a circle in (a), while the smaller velocity corresponds to the parameters denotes with a square in (a). (c) Continuation of the bifurcation diagram of (a) to larger forcing values. Note that as the forcing increases the maximum zonal flow velocity asymptotes to a constant. (d) The asymptotic zonal flow at large forcing. The collisional damping of the mean is .

Image of FIG. 5.
FIG. 5.

Structure of the eddy induced zonal flow acceleration as a function of radius. The solid line is the total flux summed over all zonal wavenumbers multiplied by 100 (at equilibrium this is equal to ). The dashed line is the acceleration induced by wavenumbers . These higher waves build the zonal flow. The dash-dotted line is the acceleration induced by the small wavenumbers which tend to destroy the zonal flow. Left panel: for mean collisional damping and the equilibrium flow in Fig. 1(d). Right panel: for (here the cancellation between downgradient and upgradient fluxes is perfect).

Image of FIG. 6.
FIG. 6.

Ratio of mean zonal kinetic energy to eddy kinetic energy as a function of stochastic excitation (solid line). Ratio of the mean zonal kinetic energy to the eddy kinetic that would have been maintained in the absence of the zonal flow (dashed line). For small excitations there is no zonal flow and the ratio vanishes, also for large excitations the flow asymptotes to a constant and again the ratio vanishes. For intermediate excitations the zonal flow energy is two to three orders of magnitude larger and the turbulence energy is dominated by the zonal flow energy. The zonal flow suppresses the eddy energy by approximately an order of magnitude. For and .

Image of FIG. 7.
FIG. 7.

(a) Structure in radius of the particle flux at equilibrium. The particle flux is not diffusive, as it has a distinct structure and there is a region of upgradient flux that would correspond to a negative diffusion coefficient. (b) The integrated particle flux at equilibrium as a function of poloidal wavenumber . (c) The structure of the eddy acceleration produced by the zonal modes. The thick solid line is the total vorticity flux which maintains the zonal flow against dissipation shown in Fig. 8. The opposing fluxes (solid and dashed lines) is the flux associated with wavenumbers while the supporting fluxes (solid and dash-dotted lines) correspond to the higher wavenumbers . (d) The energy of the eddy field as a function of poloidal wavenumber. The eddy kinetic energy peaks at the gravest zonal mode . The case is for and stochastic excitation equivalent to rms velocity of .

Image of FIG. 8.
FIG. 8.

Zonal flow at equilibrium as a function of radius. Dashed line: with no collisional damping of the mean ; solid line: with . The case is for , and stochastic forcing with equivalent rms velocity of .

Image of FIG. 9.
FIG. 9.

Top row: the top EOF of the eddy covariance of the component of the eddy field with poloidal wavenumber (on the left: perturbation electric field; on the right: perturbation density). The first EOF accounts for 32% of the total energy of the eddy field at this wavenumber. Middle row: the SO. The stochastic optimal produces 20% of the eddy energy at this wavenumber. Bottom row: the least stable eigenvalue of the operator at . The associated growth rate is . For the equilibrium zonal flow obtained at with stochastic excitation equivalent to rms velocity of .

Image of FIG. 10.
FIG. 10.

Power spectrum of the eddy energy as a function of phase speed (solid line). The dashed line is the equivalent normal response and the circles mark the maximum and minimum velocities of the equilibrium flow. (a) For . (b) For . The case is for equivalent rms velocity of and .

Image of FIG. 11.
FIG. 11.

(a) Particle flux as a function of stochastic excitation measured by equivalent ; for (solid line) and for (dashed line). (b) Maximum vorticity flux as a function of stochastic excitation. (c) Maximum equilibrium zonal flow velocity as a function of stochastic excitation; for (solid line) and for (dashed line). (d) Mean eddy kinetic energy as a function of stochastic excitation. Also shown is the eddy kinetic energy maintained against dissipation in the absence of flow as a function of stochastic excitation (dash-dotted line). For .

Image of FIG. 12.
FIG. 12.

A chaotic state (analysis of perturbed trajectory differences reveals this system to be chaotic with Lyapunov exponent ). (a) Zonal flow energy (solid line), and eddy kinetic energy (dashed line) as a function of time. (b) Zonal velocity as a function of radius and time. (c) Eddy kinetic energy, , as a function of radius and time. (d) Eddy induced zonal flow acceleration, , as a function of radius and time. The parameters are , and the stochastic excitation has equivalent rms velocity of . For these values there exists an equilibrium zonal flow with a limited basin of attraction, and this equilibrium state cannot be approached from initial states with small zonal flows.

Image of FIG. 13.
FIG. 13.

For the case shown in Fig. 12: (a) particle flux at a single location as a function of time; (b) zonal flow kinetic energy; (c) eddy kinetic energy; (d) average particle flux.

Image of FIG. 14.
FIG. 14.

A chaotic state is laminarized by impulsive introduction of a stable zonal flow at . The zonal flow subsequently asymptotically approaches the equilibrium zonal flow that exists for these parameter values. (a) Zonal velocity as a function of radius and time. (b) Zonal flow energy (solid line) and eddy kinetic energy (dashed line) as a function of time. (c) Mean particle flux as a function of time. For and .

Image of FIG. 15.
FIG. 15.

A chaotic state becomes quasiperiodic and then settles to an equilibrium as stochastic excitation increases . (a) Zonal velocity as a function of radius and time. (b) Zonal flow energy (solid line) and eddy kinetic energy (dashed line) as a function of time. (c) Mean particle flux as a function of time. (d) Stochastic excitation as a function of time. For and .

Image of FIG. 16.
FIG. 16.

Continuation of Fig. 15. The stochastic excitation is decreased to its initial value . The zonal flow persists while the eddy kinetic energy and the particle flux vanish with the excitation. (a) Zonal velocity as a function of radius and time. (b) Zonal flow energy (solid line) and eddy kinetic energy (dashed line) as a function of time. (c) Mean particle flux as a function of time. (d) Stochastic excitation as a function of time. For and .

Image of FIG. 17.
FIG. 17.

Equilibrium state diagnostics as a function of mean collisional damping. (a) Particle flux. (b) Maximum vorticity flux. (c) Maximum equilibrium zonal flow velocity. (d) Mean eddy kinetic energy. The case is for , and stochastic forcing with equivalent rms velocity of .

Image of FIG. 18.
FIG. 18.

Equilibrium state diagnostics as a function of density gradient . (a) Maximum velocity of the equilibrium zonal flow. (b) The mean particle flux (solid line). The mean particle flux increases at first linearly as (dashed line). The parameters are and the stochastic excitation supports equivalent rms velocity of .

Image of FIG. 19.
FIG. 19.

Approach to structural instability as a function of . Top: zonal flow velocities as the critical is approached. Bottom: the corresponding maximum growth rate of perturbations as a function of poloidal wavenumber . Solid line: ; dashed line: ; dash-dotted line: for . The parameters are and the stochastic excitation has equivalent rms velocity of .

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/content/aip/journal/pop/16/11/10.1063/1.3258666
2009-11-30
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: A stochastic structural stability theory model of the drift wave–zonal flow system
http://aip.metastore.ingenta.com/content/aip/journal/pop/16/11/10.1063/1.3258666
10.1063/1.3258666
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