LowestOrder Flows
To lowest order in the small parameter
, Equations (2.116) and (2.118) yield
0 

(2.132) 
0 

(2.133) 
The previous two equations can be solved to give
where
It can be seen that the lowestorder particle and heat flows are both confined to magnetic fluxsurfaces (i.e.,
).
The first term on the righthand side of Equation (2.136) is the EcrossB velocity,

(2.138) 
that is common to all plasma species [18]. The second term
is the diamagnetic velocity,

(2.139) 
which is different for electrons and ions, and is a consequence of the rapid gyromotions of charged particles in the presence of equilibrium pressure gradients [18]. The drift ordering (2.113) ensures
that the EcrossB and diamagnetic velocities are similar in magnitude. It is clear from Equation (2.137) that there is a diamagnetic flow of heat, as well as particles, around fluxsurfaces; this heat flow is the same as that
associated with the cross thermal conductivities introduced in Section 2.6.
To lowest order in the small parameter
, Equations (2.115) and (2.117) yield
Here, use has been made of
. Clearly, the
lowestorder particle and heat flows are both divergence free. Now, the fact that
implies that
. Making use of these results, the previous two equations can be combined with Equations (2.123) and (2.126), as well as the fact that
in an axisymmetric equilibrium, to give [34]

(2.142) 

(2.143) 
Taking the scalar products of Equations (2.134) and (2.135) with
, we obtain
where
Here, use has been made of Equations (2.136) and (2.137).