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we have shown that ideal pulsars calculated in the force - free limit of strong - field electrodynamics ( sfe ) dissipate a large fraction of the poynting flux in the singular current layer outside the light cylinder ( gruzinov 2011 ) .
this result  finite damping in an ideal system  is not really that unusual .
burgers equation , for instance , with viscosity @xmath0 , dissipates finite energy in infinitely thin shocks .
the standard axisymmetric pulsar magnetosphere features a nearly head - on , gruzinov ( 2005 ) ] collision of poynting fluxes right outside the light cylinder .
it is to be expected , although merely by common sense , that such a collision should be accompanied by damping .
here we show that our sfe solution also obtains from the standard force - free magnetosphere equation of scharlemant & wagoner ( 1973 ) , if one uses the `` correct '' boundary condition at the equatorial current layer .
we propose that the `` correct '' boundary condition at the singular current layer ( which now exists only outside the light cylinder ) is [ bc ] b^2-e^2=0 .
this condition is lorentz invariant , comes up in the sfe simulations , and has a clear physical meaning ( at equator , the field becomes electric - like in order to drive large current )
.    we can not be sure that our proposal works , until one justifies the full sfe , or just eq.([bc ] ) , microscopically .
but conversion of 50% of the poynting flux into the ohmic power ( radiation , electron - positron pairs ) occurring close to the light cylinder must have consequences for the pulsar phenomenology , and needs to be studied .    in  2
we derive the pulsar magnetosphere equation and explain how contopoulos , kazanas & fendt ( 1999 ) solve it . in 
3 we put together all the equations which are needed to calculate the pulsar magnetosphere . in 
4 we describe the numerical solution and the corresponding physics results .
goldreich & julian ( 1969 ) proposed that neutron star magnetospheres obey the force - free condition [ ff ] + * j*=0 .
surprisingly , it turns out that this simple equations allows a full calculation of the pulsar magnetosphere ( scharlemant & wagoner 1973 , contopoulos , kazanas & fendt 1999 , gruzinov 2005 , spitkovsky 2006 ) .    for the stationary axisymmetric case ,
the calculation is as follows . using axisymmetry and stationarity , in cylindrical coordinates
@xmath1 , we represent the fields by the three scalars @xmath2 , @xmath3 , and @xmath4 , which depend on @xmath5 and @xmath6 but not on @xmath7 : [ stat ] * e * = - ,   * b*=1r(-_z , a,_r ) , where the subscripts denote the partial derivatives .
we plug ( [ stat ] ) into ( [ ff ] ) and use @xmath8 and @xmath9 .
we also use the boundary conditions at the surface of the star  the continuity of the normal component of the magnetic field and the tangential component of the electric field .
we use the pulsar units = = c=1 , where @xmath10 is the magnetic dipole moment of the star .
it is assumed that the magnetic field is a pure dipole near the surface inside the star .
the star is assumed to be a perfect conductor .
@xmath11 is the angular velocity of the star .
we get = ,   a = a ( ) , where @xmath4 is an arbitrary function of @xmath3 , and we also get the `` grad - shafranov - like '' pulsar magnetosphere equation for @xmath3 [ pme ] ( 1-r^2)-2r_r+f()=0 . here
@xmath12 , and @xmath13 , where the prime denotes the @xmath3-derivative . the pulsar magnetosphere equation ( [ pme ] )
is solved outside the star r^2+z^2>r_s^2 , with the boundary condition at the surface of the star = r^2r_s^3 ,   r^2+z^2=r_s^2 .
the pulsar magnetosphere equation ( [ pme ] ) contains @xmath14  an arbitrary function of @xmath3 , and it is not clear how one should solve it .
this was explained and done by contopoulos , kazanas & fendt ( 1999 ) ( ckf ) .
the pulsar magnetosphere equation is elliptical both inside and outside the light cylinder , and can therefore be solved if some conditions are given at all boundaries and if @xmath14 is known .
we first pick a trial @xmath14 .    then inside the light cylinder
, we have all the boundary conditions : ( i ) we know @xmath3 at the surface of the star , ( ii ) @xmath15 at @xmath16 , ( iii ) at @xmath17 the boundary condition is given by the pulsar magnetosphere equation itself , @xmath18 .
so we can find @xmath3 inside the light cylinder , say by the relaxation method using the variation principle .    outside the light cylinder ,
ckf postulate the boundary condition at the equator , ( r,0)=(1,0 ) ,  r>1 . then , with some boundary conditions at infinity ( we use @xmath15 at @xmath16 and @xmath19 at @xmath20 ) , and with the same boundary condition at the light cylinder , @xmath18 at @xmath17 , one can solve the pulsar magnetosphere equation outside the light cylinder too .    for a generic @xmath14 , this procedure gives a solution with @xmath21 .
but one might hope that there is a ( unique ? )
@xmath14 which gives a smooth solution .
ckf use a feedback procedure  numerical adjustment of @xmath14 leading to a smooth light cylinder crossing .    in our case ,
the boundary condition at the equator outside the light cylinder is ( r^2 - 1)()^2=a^2 ,  z=0 ,  r>1 . at the light cylinder , @xmath17
, this gives a(_0)=0 ,  _
0(1,0 ) , meaning that there is no singular return current on the field line @xmath22 .
the only singularity is the equatorial current layer .
in summary , we solve the following pulsar magnetosphere equation [ pme1 ] ( 1-r^2)-2r_r+aa=0 , [ pme2 ] = r^2r_s^3 ,   r^2+z^2=r_s^2 , [ pme3 ] ( r^2 - 1)()^2=a^2 ,  z=0 ,  r>1 .
the function @xmath23 is ( uniquely ? ) determined by the continuity at the light cylinder .
the poynting power is @xmath24 .
our numerical procedure is as follows .
we pick a trial function g(r)>0 ,   g(r)<0 ,  r>1 ,  g(1)=1 . instead of the boundary condition ( [ pme3 ] ) , we impose ( r,0)=_0g(r ) ,   r>1 ,   _ 0(1,0 ) .
we apply the ckf relaxation method to calculate @xmath25 ( ignoring the singular return current ) .
then we calculate the field invariant at the equator outside the light cylinder i(r)=1r^2(a^2+()^2)-()^2 ,  z=0 ,  r>1 .
we then _ manually _ chose @xmath26 , so as to make @xmath27 as close to zero as we can at all @xmath28 .
it turns out , that g(r)=0.52 + 0.48r , nullifies @xmath27 to about the numerical accuracy .
has been read off the sfe time - dependent simulation ( we then confirm that various other profiles @xmath26 do nt nullify @xmath27 to the accuracy shown in fig.1 . ) . what one really wants is a ckf - type feedback loop , which would adjust @xmath26 so as to nullify @xmath27 . or
, maybe , one can enforce the correct boundary condition ( [ pme3 ] ) throughout the @xmath14 relaxation .
we were unable to develop a numerical scheme which would solve the problem ( [ pme1 ] - [ pme3 ] ) all by itself . ]
one gets the spin - down power ( the poynting flux at the stellar surface = the poynting flux through the light cylinder ) : l_sd 0.9 ^ 2 ^4c^3 . the ohmic power ( the poynting flux on the field lines that cross the equatorial current layer ) is l_ohm 0.5l_sd . the ohmic power between 1 and 1.5 light cylinder radii is l_ohm 1.5 0.1l_sd .
ideal axisymmetric pulsar magnetosphere is calculated from the standard stationary force - free equation but with a new boundary condition at the equator . 
 the new solution predicts ohmic heating . about 50% of the poynting power 
 is dissipated in the equatorial current layer outside the light cylinder , with about 10% dissipated between 1 and 1.5 light cylinder radii . 
 the ohmic heat presumably goes into radiation , pair production , and acceleration of charges  in an unknown proportion .