text: The theory is based on the principle that the statistical properties are invariant for rotations about a particular direction  (say), and reflections in planes containing  and perpendicular to . This type of axisymmetry is sometimes referred to as strong axisymmetry or axisymmetry in the strong sense, opposed to weak axisymmetry,  where reflections in planes perpendicular to  or planes containing  are not allowed.
  Let the two-point correlation for homogeneous turbulence be 
  A single scalar describes this correlation tensor in isotropic turbulence, whereas, it turns out for axisymmetric turbulence, two scalar functions are enough to uniquely specify the correlation tensor. In fact, Batchelor was unable to express the correlation tensor in terms of two scalar functions, but ended up with four scalar functions, nevertheless, Chandrasekhar showed that it could be expressed with only two scalar functions by expressing the solenoidal axisymmetric tensor as the curl of a general axisymmetric skew tensor (reflectionally non-invariant tensor).
  Let  be the unit vector which defines the axis of symmetry of the flow, then we have two scalar variables,  and . Since , it is clear that  represents the cosine of the angle between  and . Let  and  be the two scalar functions that describes the correlation function, then the most general axisymmetric tensor which is solenoidal (incompressible) is given by,
  where
  The differential operators appearing in the above expressions are defined as
  Then the evolution equations (equivalent form of Kármán–Howarth equation) for the two scalar functions are given by
  where  is the kinematic viscosity and 
  The scalar functions  and  are related to triply correlated tensor , exactly the same way  and  are related to the two point correlated tensor . The triply correlated tensor is
  Here  is the density of the fluid.