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Thus, in 3 dimensions, the shear is a 3 x 3 tensor.
Thus, in 3 dimensions, the shear is a 3 x 3 tensor.


In the context of magnetic confinement, and assuming the existence of toroidally nested magnetic [[Flux surface|flux surfaces]], the only relevant variation of the direction of the magnetic field is the radial gradient of the '''rotational transform''' (field line pitch). The latter is defined as
== Global magnetic shear ==


:<math>\frac{\iota}{2 \pi} = \frac{d \psi}{d \phi}</math>
In the context of magnetic confinement, and assuming the existence of toroidally nested magnetic [[Flux surface|flux surfaces]], the only relevant variation of the direction of the magnetic field is the radial gradient of the [[Rotational transform|rotational transform]].
 
The global magnetic shear is defined as
where ''&psi;'' is the poloidal magnetic flux, and ''&phi;'' the toroidal magnetic flux.
Thus, ''&iota;/2&pi;'' is the mean number of toroidal transits (''n'') divided by the
mean number of poloidal transits (''m'') of a field line on a flux surface.
In tokamak research, the quantity ''q = 2&pi;/&iota;'' is preferred (called the "safety factor").
 
The (global) magnetic shear is defined as


:<math>s = \frac{r}{q} \frac{dq}{dr} = -\frac{r}{\iota} \frac{d\iota}{dr}</math>
:<math>s = \frac{r}{q} \frac{dq}{dr} = -\frac{r}{\iota} \frac{d\iota}{dr}</math>
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High values of magnetic shear provide stability, since the radial extension of helically resonant modes is reduced.
High values of magnetic shear provide stability, since the radial extension of helically resonant modes is reduced.
Negative shear also provides stability, possibly because convective cells, generated by curvature-driven instabilities, are sheared apart as the field lines twist around the torus.
Negative shear also provides stability, possibly because convective cells, generated by curvature-driven instabilities, are sheared apart as the field lines twist around the torus.
<ref>[http://link.aip.org/link/?PHPAEN/3/2221/1 T.M. Antonsen, Jr., et al, ''Physical mechanism of enhanced stability from negative shear in tokamaks: Implications for edge transport and the L-H transition'', Phys. Plasmas '''3''', 2221 (1996)]</ref>
<ref>T.M. Antonsen, Jr., et al, ''Physical mechanism of enhanced stability from negative shear in tokamaks: Implications for edge transport and the L-H transition'', [[doi:10.1063/1.871928|Phys. Plasmas '''3''', 2221 (1996)]]</ref>
 
== Local magnetic shear ==


The (local) magnetic shear is defined as
The local magnetic shear is defined as
<ref>[http://link.aip.org/link/?PHPAEN/8/4375/1 M. Nadeem et al, ''Local magnetic shear and drift waves in stellarators'', Phys. Plasmas '''8''' (2001) 4375]</ref>
<ref>M. Nadeem et al, ''Local magnetic shear and drift waves in stellarators'', [[doi:10.1063/1.1396842|Phys. Plasmas '''8''' (2001) 4375]]</ref>


:<math>s_{\rm local} = 2 \pi \vec{h} \cdot \vec{\nabla} \times \vec{h}</math>
:<math>s_{\rm local} = 2 \pi \vec{h} \cdot \vec{\nabla} \times \vec{h}</math>
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where  
where  


:<math>\vec{h} = \frac{\vec{\nabla} \psi}{|\vec{\nabla} \psi|} \times \vec{b}</math>
:<math>\vec{h} = \frac{\vec{\nabla} \psi}{|\vec{\nabla} \psi|} \times \frac{\vec{B}}{|\vec{B}|}</math>
 
== See also ==


and
* [[Rotational transform]]
* [[Connection length]]


:<math>\vec{b} = \frac{\vec{B}}{|\vec{B}|}</math>
== References ==
== References ==
<references />
<references />
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