Abstract
A remarkable phenomenon in turbulent flows is the spontaneous emergence of coherent large spatial scale zonal jets. In this work a comprehensive theory for the interaction of jets with turbulence, stochastic structural stability theory, is applied to the problem of understanding the formation and maintenance of the zonal jets that are crucial for enhancing plasma confinement in fusion devices.
Discussions with S. Nazarenko and B. Nadiga are gratefully acknowledged. This work was supported by NSF Grant No. ATM0123389.
I. INTRODUCTION
II. FORMULATION
A. The HW drift wave turbulence equations
B. The SSST system governing DWZF dynamics
C. Parameters
III. DWZF BEHAVIOR IN PARTICULAR REGIMES OF DYNAMICAL INTEREST
A. Formation of zonal jets starting from a nonequilibrium state
B. Structural instability of the zero zonal flow state as a function of the amplitude of the stochastic excitation in the absence of drift wave instability,
C. Zonal flow equilibria as a function of the amplitude of stochastic excitation in the presence of drift wave instability,
D. Zonal flow equilibria for
E. Loss of structural stability at large
IV. DISCUSSION
V. CONCLUSION
Key Topics
 Zonal flows
 91.0
 Turbulent flows
 79.0
 Eddies
 56.0
 Flow instabilities
 24.0
 Turbulent jets
 19.0
Figures
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.
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.
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 (dashdotted 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 .
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 (dashdotted 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 .
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.
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.
(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 .
(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 .
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 dashdotted 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).
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 dashdotted 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).
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 .
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 .
(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 dashdotted 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 .
(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 dashdotted 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 .
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 .
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 .
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 .
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 .
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 .
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 .
(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 (dashdotted line). For .
(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 (dashdotted line). For .
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.
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.
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.
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.
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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: ; dashdotted line: for . The parameters are and the stochastic excitation has equivalent rms velocity of .
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: ; dashdotted line: for . The parameters are and the stochastic excitation has equivalent rms velocity of .
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