Schematic illustration of the flow pattern in a tokamak poloidal plane. A simplified calculation is presented here, where the fluid parallel velocity and its gradient are small. In other words only the residual stress tensor is calculated. It appears that the sign of the correlation between the poloidal and parallel components of the Reynolds stress translates in a correlation of the curvature and drift contributions to the radial flux of parallel momentum. This general result comes from a simple relationship between the momentum flux due to curvature and the turbulent Reynolds stress summarized in figure 2.
Although poloidal convective cells do not appear explicitly in this relationship, they are instrumental to this mechanism. The overall process is illustrated by a quasi-linear calculation of stress tensors and fluxes based on linear ion temperature gradient ITG driven modes in the hydrodynamic limit. This calculation predicts similar amplitudes for the and magnetic drift components of the parallel momentum flux, but a sign of correlation that depends on plasma parameters.
Typically for modes drifting in the ion diamagnetic direction, positive correlation is found for weak drive, and anti-correlation for strong drive. Since the hydrodynamic limit is valid only in the strong drive limit, this result can be considered as encouraging in view of previous numerical calculations, which found anti-correlation [ 13 , 16 ]. The paper is organized as follows.
General expressions of radial fluxes of parallel momentum are derived in section 2. The contribution from the magnetic drift is detailed in section 3. The and curvature driven fluxes of parallel momentum are compared and discussed in section 4. A conclusion follows. Most technical demonstrations can be found in the appendices, and may be skipped on first reading.
Momentum flux is the same as the Reynolds stress tensor up to a mass density Nm , where N is the unperturbed density and m the ion mass.
Both names will be used indistinctly throughout the paper. The analysis is restricted to electrostatic turbulence, i. A simplified geometry of circular concentric magnetic surfaces is considered here. The inverse aspect ratio is a small parameter throughout the paper. Since an evaluation of the turbulent Reynolds stress tensor is needed, it is necessary to postulate a spatial structure of the fluctuations of the electric potential. We follow a rationale that is close in spirit to the ballooning representation [ 24 — 29 ].
In a magnetized plasma with strong guide field, turbulent structures tend to align with the magnetic field. Given the structure of the magnetic field equation 5 , it is convenient to use the variable instead of the toroidal angle. The periodicity in the toroidal direction allows using a a Fourier series. The periodicity condition in the poloidal direction imposes. Separability is not always rigorously demonstrated, but is a good proxy.
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The amplitude is called envelope, usually a localized around an angle called ballooning angle, with a narrow width of the order of , where L p is a gradient length and the ion thermal gyroradius radius. The ballooning angle measures the departure from up—down symmetry on a given field line. Non zero values of result from symmetry breaking mechanisms, as will be seen later on. This leads to a slow radial dependence of in r. Hence 'fast' radial variations of come from the radial dependence of the safety factor in the exponential argument of equation 6 , with a characteristic scale , whereas 'slow' radial variations come from the amplitude , with a mesoscale length.
We insist here on the fact that a 'slow' radial scale is not a mean gradient length—the only constraint is that it should be larger than the 'fast' scale. Calculating a mode envelope in turbulent regime is questionable since a global mode has presumably no enough time to form before turbulent decorrelation occurs, except very close to the threshold. Nevertheless simulations show that an instantaneous ballooning angle can still be identified [ 16 ]. We therefore adopt the following representation in non linear regime.
The vector is a label for the couple so that the sum over designates a summation over the toroidal wave number n and also an integral over a distribution of angles. The value of the angle depends on time, as shown by simulations [ 16 ]. To some extent, this time dependence can be represented by such a distribution. Quantities labeled with a index will be called 'Fourier' harmonics, since can be seen as the Fourier counterpart of the radial variable.
Each harmonic depends also slowly on the radius r. All quantities depends on with a typical scale that is larger than a typical turbulent vortex size, but is smaller than a gradient length. The label is omitted to simplify the notations, unless specified otherwise. Nevertheless it is a reasonable proxy for a turbulence localized on the low field side of a tokamak i. Hence each Fourier component contains the information on the poloidal localization of fluctuations, and therefore poloidal asymmetries. We postulate a Gaussian form for i. The value of the ballooning angle is not easy to determine.
In the ballooning formalism, it comes from a calculation of the mode envelope. However, as already mentioned, this calculation is questionable for a turbulent state. An estimate can be found by using a rapid distortion theory [ 30 ]. Indeed a structure with initially a zero ballooning angle will acquire an effective radial wave number due to a shear flow that is equal to after a time t.
Here is the shear flow rate, i. Defining a correlation time , one gets the following estimate. Other expressions of have been given in the past [ 26 — 29 , 31 ], with similar structures as equation 10 when envelopes are spatially localized. The exact expression of has no real importance here, since we are interested in the relative signs and amplitudes of the momentum fluxes.
Sources of symmetry breaking different from shear flow will lead to different expressions of , but will cause correlated radial and parallel wavenumbers. We estimate now the radial fluxes of poloidal and parallel momentum due to drift, which are proportional to the corresponding components of the Reynolds stress. The equations equations 11 , 12 can then be recast as. The wavenumber is a reference poloidal wave number defined as. The poloidal structure function is defined as see details in appendix A. The amplitude does not depend on time for the structure equation 8.
Note that at this stage, the fluxes depend on the poloidal angle and time plus a slow radial variation in. A rough estimate of is obtained by a rapid distortion argument [ 30 ], i. The components of the Reynolds stress can be expanded as Fourier series in the poloidal direction. Using the expressions equations 23 , 24 , one finds.
The coefficients and measure the distortion of turbulent structures in the radial and parallel directions. They can be formally written as see appendix A. The wave number characterizes fluctuation poloidal symmetries. The useful values of are low, typically , and therefore different from the much larger turbulent poloidal wave numbers.
The coefficients and can be seen as effective radial and parallel wavenumbers for a given set. Calculations in appendix A show that the coefficients and are close to their components for small ballooning angles , i. These coefficients are related to the parameters used for the turbulent field representation. The symbol resp. In view of the field structure equation 9 , the real part of must be positive, i. As expected, the average radial and parallel wave numbers are correlated and proportional to , which measures the strength of the symmetry breaking mechanism.
However a close inspection of equations 28 , 29 indicates that the proportionality coefficients are not the same. For instance, in the limit non propagative localized mode , it appears that is of the sign of , while is of the sign of. Hence the sign and amplitude of the correlation between the radial and parallel wave numbers is by no way trivial. This relationship has been discussed in [ 32 ], in the context of turbulent momentum transport in slab geometry.
Here calculations are restricted to order one in.
Impact of poloidal convective cells on momentum flux in tokamaks - IOPscience
Since we are interested in signs, a rapid distortion theory may not be accurate enough. A quasi-linear theory provides a more precise value of the time and poloidal average of. The calculation is done in appendix B , and confirms the estimate equation The structure of the later is in line with previous calculations of the residual stress see [ 33 — 35 ], and overviews [ 19 , 21 ]. The structure of the flux of poloidal momentum equation 21 is well-known [ 36 ] and was used abundantly in the context of transport barrier formation.
We note that and are anti-correlated when and are of the same sign. The expression equation 2 of the radial flux of parallel momentum due to magnetic drift shows that a finite flux can only be due to up—down symmetries of the distribution function. The rationale here is as follows: because turbulence is ballooned, the stress tensor is ballooned too, thus leading to poloidal asymmetries of the poloidal flow, and therefore of the electric potential. The resulting distorted distribution function is correlated with the Reynolds stress, and therefore with the radial flux of parallel momentum.
Damping of time dependent and poloidally asymmetric flows is not primarily due to collisions, but rather wave particle resonant effects. An estimate can be obtained by solving the linear gyrokinetic equation see [ 37 , 38 ] for overviews on gyrokinetic theory. The parallel non linear term has been neglected.
The mirror force is also ignored since the quantity of interest is the ion parallel flux, to which passing particles mostly contribute. This nevertheless sets a lower bound in frequency, namely the trapped ion bounce frequency of the order of. It is reminded that the inverse aspect ratio is a small parameter in the present work. To simplify the calculation, we use the long wavelength limit of the gyroaverage operator , i. For non zonal modes , this gives an equation for the potential vorticity , i. The vorticity equation 33 has been derived by assuming that the electric potential wavelength is smaller than the gradient lengths of density and temperature.
One can show the following variant of the Taylor identity [ 39 ] see appendix C. It is reminded that is the reference radius in the neighborhood of which the calculations are performed. Keeping FLR effects would replace one component of the drift velocity by in the stress tensor , i. This does not change the structure of the calculation as long as pressure fluctuations are proportional to potential fluctuations. Implications are discussed in section 4.
A subsidiary small parameter is introduced at this point, which is the ratio of the thermal Larmor radius to the typical radial wavelength of the turbulent Reynolds stress tensor. Only terms of order are retained in the following. The divergence of the parallel flux in the vorticity equation 33 can be calculated by computing the axisymmetric components of the perturbed distribution function, which is a linear solution of the gyrokinetic equation 31 , i. It appears that the component of the parallel flux divergence is. The explicit expressions of and are.
The last term in the vorticity equation 33 cancels out since the divergence of the diamagnetic current term is linear so that its toroidal average is zero. The other term is a particle flux which is equal to zero for an ITG turbulence. The vorticity equation 33 can then be solved by combining the Taylor identity equation 35 with the parallel flow divergence equation 38 , thus providing the Fourier components of the electric potential.
It is reminded that only the leading term in is kept. The expression of the electric potential Fourier components equation 41 is of central importance since it provides the structure of time-dependent poloidal convective cells which are driven by turbulent eddies. This result calls for several comments:. Therefore one is left with the cross-correlation between the magnetic drift and the perturbed distribution function due to the electric potential fluctuations driven by turbulent eddies.
This cross-correlation is responsible for a finite flux of parallel momentum. The time dependent radial flux of parallel momentum due to magnetic drift reads. This flux can be calculated using the expression equation 41 of the amplitudes of poloidal convective cells see appendix D. A compact expression is found for its time average.
For a ballooned turbulence, the Reynolds stress is ballooned too. If fluctuations are strongly localized on the low field side, the prefactor can be replaced by 1 in equation This can also be demonstrated by using the structure of the stress tensor equation 21 and the expression of with see equation Given the simplicity of equation 44 , one may actually wonder whether a more direct calculation of is possible, and indeed it is.
Since the parallel velocity plays a subdominant role in the magnetic drift, essentially because the resonant velocity goes like the pulsation, the curvature driven flux of momentum equation 43 can be reformulated as. An integration by part allows a reformulation in terms of the parallel gradient of the parallel flux. Using the vorticity equation 33 , and the Taylor identity equation 35 , one finds equation It is quite remarkable that this result does not depend on the details of the poloidal convective cells, which act as mediators. An example of turbulence self-organization via the generation of poloidal convective cells is shown in figure 3.
It turns out that poloidal convective cells play also a role in the interplay between turbulent and neoclassical impurity transport [ 17 ].
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Figure 3. The expression equation 44 of the radial flux of parallel momentum due to the magnetic drift can be expressed in the Fourier space using equation Banks, O. Barnouin, V. Bray, B. Carcich, A. Chaikin, C. Chavez, C. Conrad, D. Hamilton, C. Howett, J. Hofgartner, J. Kammer, C. Lisse, A. Marcotte, A. Parker, K. Retherford, M. Saina, K. Runyon, E. Schindhelm, J. Stansberry, A. Steffl, T.
Stryk, H. Throop, C. Tsang, A. Verbiscer, H. Winters, A. Zangari, S. Nature , ; : 82 DOI: ScienceDaily, 6 June Washington University in St.
Pluto as a cosmic lava lamp: Giant convective cells continually refresh dwarf planet's icy heart. Retrieved September 21, from www. Of particular interest is the heart's The idea that Pluto has a Astronomers show that Pluto's peculiar insolation and atmosphere favor nitrogen In this process the warm air is cooled; it gains density and falls towards the earth and the cell repeats the cycle.
Convection cells can form in any fluid, including the Earth's atmosphere where they are called Hadley cells , boiling water, soup where the cells can be identified by the particles they transport, such as grains of rice , the ocean, or the surface of the sun. The size of convection cells is largely determined by the fluid's properties. Convection cells can even occur when the heating of a fluid is uniform. A rising body of fluid typically loses heat when it encounters a cold surface when it exchanges heat with colder liquid through direct exchange, or in the example of the Earth's atmosphere , when it radiates heat.
At some point, the fluid becomes denser than the fluid beneath it, which is still rising. Since it cannot descend through the rising fluid, it moves to one side. At some distance, its downward force overcomes the rising force beneath it, and the fluid begins to descend.
go As it descends, it warms again through surface contact or conductivity and the cycle repeats. Warm air has a lower density than cool air, so warm air rises within cooler air,  similar to hot air balloons. As the moist air rises, it cools, causing some of the water vapor in the rising packet of air to condense. If enough instability is present in the atmosphere, this process will continue long enough for cumulonimbus clouds to form, which support lightning and thunder. Generally, thunderstorms require three conditions to form: moisture, an unstable air mass, and a lifting force heat.
All thunderstorms, regardless of type, go through three stages: a 'developing stage', a 'mature stage', and a 'dissipating stage'. The plasma cools as it rises and descends in the narrow spaces between the granules.
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