The collision of plates is often accompanied by earthquakes and volcanoes. The teachers were asked to sketch the variation in the distribution of heat from the equator to the poles, noting the difference in the angle of incidence with latitude and how this would affect heating. One would expect to see warmer temperatures at the equator and cooler temperatures at the poles leading to two large convection cells from the equator to the poles, one in each hemisphere.
The teachers pondered this and other questions to be addressed further during Session 2 on November 2, 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.
From Wikipedia, the free encyclopedia. See also: Convection. See also: Cloud and Thunderstorm. Gaponenko and V.
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Mogil An instructive example is the case of an unstable linear toroidal ITG driven mode. General expressions of and are derived in the hydrodynamic limit in the appendix E. In the long wavelength limit and strong magnetic shear , explicit expressions of the coefficients and can be found.
The numbers and depend on the normalized frequencies and only, where and are the real and imaginary parts of the linear frequency in the plasma frame frame where the local radial electric field is zero , and. The coefficient is always positive. These expressions are valid for any sign of and.
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We now restrict the discussion to the generic case and. These considerations lead to the following estimates of the momentum fluxes. For , the two numbers and are of the same order of magnitude. The two fluxes are therefore comparable when.
This condition appears as reasonable since the intensity wave number spectrum typically peaks at , provided [ 44 , 45 ]. Regarding the signs, it appears that for modes that drift in the ion diamagnetic direction , is negative for a weak drive , and positive for strong drive. Since is always positive, this means that and are positively correlated near threshold, and anti-correlated far from threshold. Consequently the two fluxes or parallel momentum are positively correlated for low drive and anti-correlated for strong drive.
For modes drifting in the electron diamagnetic direction, , anti-correlation always occurs. For ITG modes, drift in the ion diamagnetic direction is expected, so that anti-correlation is expected only far enough from the instability threshold. It is not clear whether this finding agrees or not with previous numerical findings for momentum transport [ 13 , 16 ]. Nevertheless it is stressed that the hydrodynamic limit that is being used here, is a rather poor approximation that becomes correct only well above the instability threshold, i.
This is precisely the regime where an agreement with simulations is found, i. In that regard some further analysis of the numerical simulations would be helpful. An encouraging observation though [ 16 ] is that the flux of parallel momentum is anti-correlated with the ballooning angle , which suggests positive in view of equation Let us note that adding the FLR effects mentioned in section 3.
Indeed changing the stress tensor into is roughly equivalent to multiplying in equation 52 by a factor , where is the pressure diamagnetic frequency. As a final note, it is possible if not likely that the two terms and are determined by non linear processes and not well captured by a linear analysis. It is shown here that the Reynolds stress tensor generates poloidal asymmetries of the plasma flow due to turbulence ballooning.
These poloidal convective cells are weakly damped at low frequency. Their radial scale is dictated by the turbulent Reynolds stress, and their poloidal wave numbers are small.
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These cells drive up—down asymmetries of the distribution function, which are responsible for a non-zero radial flux of parallel momentum due to the geodesic component of the particle magnetic drift. The entire process requires a symmetry breaking mechanism, for instance a mean shear flow. Since the turbulent Reynolds can be seen as a flux of momentum, it appears that the two components of the radial flux of parallel momentum due to curvature and drift are correlated.
This general result comes from a simple relationship between the momentum flux due to magnetic drift and the turbulent Reynolds stress.
Although poloidal convective cells do not appear explicitly in this relationship, they play an essential role in this mechanism. An analytic calculation shows that anti-correlation between the components of the turbulent Reynolds stress results in anti-correlation of the two contributions to the flux of parallel momentum that come from and magnetic drifts. A quasi-linear calculation of all quantities, based on ITG linear stability indicates that positive correlation is expected near threshold, and anti-correlation for strong drive. Hence no firm conclusion can be drawn as to the relevance of this mechanism to explain the numerical results.
Nevertheless the hydrodynamic limit that has been used is a rather poor representation of the turbulence near threshold. In fact it is valid only well above the stability threshold, i. This is encouraging since it is the case where anti-correlation is found, in agreement with numerical findings. It is likely that poloidal convective cells generated by turbulence have other consequences on turbulent transport and turbulence.
Indeed they may participate in turbulence self-regulation via vortex shearing processes similar to the well documented effect of zonal flows on turbulence. The mode structure equation 9 lead to a turbulence intensity that reads. The amplitude in equation 9 has been chosen such that. The poloidal Fourier components of the radial and parallel wave numbers equation 25 and equation 26 can then be recast as.
For a strongly ballooned turbulence , and small values of the ballooning angle , one gets at first order in.
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Hee and indicate the real and imaginary parts of a complex number z. To calculate the distribution function versus the potential, we use a ballooning representation. The electric potential is written in the form.
A similar expansion is used for the non adiabatic part of the distribution function, i. The gyrokinetic equation 31 reads. Here is an operator.
The kinetic magnetic drift frequency is defined as. The definition of the kinetic diamagnetic frequency is the usual one. The gyroaverage is fairly well represented by a Bessel function with an argument , i. The solution of the gyrokinetic equation 67 involves an integro-differential operator that relates to [ 46 , 47 ].
A formal solution can be written in a Wentzel-Kramers-Brilloin WKB sense by dividing the rhs of equation 67 by the resonant term. In the hydrodynamic limit , one gets the result. This result is identical to equation 22 with. Strictly speaking there is also a contribution to the flux of parallel momentum that comes from the axisymmetric perturbations of the potential equation 41 and the distribution function response equation However this contribution is of second order in ballooning angle and will not be retained here. Hence the flux of momentum is in this peculiar case 'turbulent', i.
The demonstration is restricted to a geometry of concentric circular surfaces with large aspect ratio.