Beta in TRANSP. ============== (This note updated August 2007, with description of Li_1 and Li_3 normalized inductances added to TRANSP output at that time-- for details see the new appendix at the end of document). The TRANSP Beta definitions displayed in RPLOT take the general forms: Beta = P / ( /8*pi ) (toroidal beta) or Beta = P / ( /8*pi ) (poloidal beta) or Beta = P / ( (mu0*Ip/2*pi*a)**2/8*pi ) (1D poloidal beta) Ip=plasma current a=midplane halfwidth Please note that the averages <> of B**2 can be subtle, because TRANSP does not employ concentric-circular flux surface geometric approximations. In particular: is the VOLUME AVERAGE over the entire plasma volume of the VACUUM toroidal field strength squared. Neither poloidal field corrections nor para/diamagnetic corrections are applied. is the differential volume average of the poloidal field strength squared, Bp**2, over the outermost flux surface. Note that this normal- ization can differ SIGNIFICANTLY from 1D approximations e.g. taking Bp=mu0*Ip/2*pi*a, with a as the midplane halfwidth of the plasma. Therefore, the 1d Beta is also output for comparison. Differences arise not only due to non-circular flux surface shapes but also from the degree of flux surface shifting which effects the magnitude of Bp going around the flux surface. The definition of the numerator P is given now for various plasma components: Thermal: P = n*T n = particle density, T = temperature = 0.666667*thermal plasma energy density. Rotation: P = 1.333333*rotational energy density i*omega**2/2 i=angular inertia omega=angular velocity Fast ion perp: P = fast ion perpendicular energy density Fast ion parallel: P = 2*fast ion parallel energy density (Beta components are broken down by fast ion species-- e.g. beams, RF, fusion products) and the total equilibrium beta is given by Beta(eq) = Rotation beta + sum(all thermal betas) + 1/2 * [sum(all fast ion perp betas) + sum(all fast ion parallel betas)] and the total diamagnetic beta is given by Beta(dia)= sum(all thermal betas) + sum(all fast ion perp betas) In the TRANSP output there are also fast ion *toroidal* beta profiles which are sums over perpendicular and parallel components of the form (e.g. for beam ions) Beta(beam-toroidal) = 2/3*[Beta(beam-toroidal-perp) + 1/2 * Beta(beam-toroidal-pll)] = 2/3*[total beam energy density]/( /8*pi ) The choice of these definitions appears to be entirely historical. ------------------------------------------------------------------ Equilibrium Pressure in TRANSP. =============================== (dmc 25 Jan 1994): definitions of functions in the PRESS multigraph (PLASMA PRESSURE, PASCALS, vs. time and flux coordinate). Due to repeated inquiries, I give these definitions here. These definitions may be recomputed and checked in the RPLOT calculator: PPLAS = 0.6666667E6 * ( UE + UI ) UE and UI are the electron and ion thermal energy densities, JLES/CM3, respectively. UE=1.5*1.602E-19*NE*TE, UI=1.5*1.602E-19*NI*TI, where NE, NI are plasma electron and ion densities, CM**-3, and TE, Ti are plasma electron and ion temperatures, eV. PTOWB = PPLAS + (1.333333*UPHI + UFASTPA + 0.5*UFASTPP)*1.0E6 UPHI is the rotation energy density, UFASTPA is the total fast ion parallel energy density (summed over fast ion species), and UFASTPP is the total fast ion perpendicular energy density (summed over fast ion species). All in JLES/CM3. Note that the PTOWB is supposed to be an "MHD equilibrium" pressure definition, more related to the Beta(eq) definition given in the previous section than to the various pressures P and component beta definitions given above. NOTE dmc 20 Aug 1998 -- I have added a namelist factor XUPHIFAC, which has a default value of 1.333333, which now becomes the factor in front of UPHI for evaluation of PTOWB and the rotational "beta" and "pressure". NOTE dmc 6 Apr 2004 -- various new TRANSP options allow the equilibrium pressure to be adjusted before being fed to the equilibrium solver. Thus PMHD_IN = PTOWB with adjustments. In most cases, PMHD_IN = PTOWB, but exceptions are created as needed in practice. For example, in recent JET current hole experiments, the profile in PMHD_IN is flattened in the core region where there is no current, as is necessary to obtain a solution. It is also possible for the entire PMHD_IN profile to be adjusted to meet some other data constraint such as matching a measurement of li/2+beta. ------------------------------------------------------------------ editorial comments: These definitions were verified in TRANSP in 25 Jan 1993 and have not changed. The definitions appear to be based on: L.L Lao, H. St. John, R.D. Stambaugh, & W. Pfeiffer, Nucl. Fus <25>, 1421-1436 (1985). but the definitions are not universally accepted. Over the years I have received many questions / complaints about definitions of Betas and pressures output by TRANSP. The problem here is that different people have expected different definitions of these quantities, and a clear consensus of what constitute the "correct" definitions does not seem to exist. I believe that the energy density outputs of TRANSP are correct and unambiguous. These are the quantities contained in the RPLOT multigraph package UDENS, units JLES/CM3. To form various definitions of plasma pressure, I recommend using various combinations of the UDENS functions in the RPLOT calculator, or in a postprocessing program accessing TRANSP data e.g. with the TRREAD module in the NTCC modules library, cf http://w3.pppl.gov/NTCC. I recommend entirely avoiding the use of "Beta" in scientific communications, because this frequently leads to arguments over definitions later on. ............................... Douglas McCune ... 1988 ... reviewed 1/1994 ... updated 2/2002 ... updated 8/2007 ----------------------------------------------------------------------------- Appendix: LI_1, LI_3, LI_VDIFF; LINORM scalar multigraph In August 2007, the scalar multigraph LINORM was added to the TRANSP output. This multigraph includes various ITER standard inductance definitions: LI_1 = volint(Bpol**2)/(Pvol*(mu0*Ip/Lpol)**2) LI_3 = volint(Bpol**2)/((mu0*Ip)**2*RLI_3) LI_VDIFF = volint(Bpol**2)/(Pvol*<|Bpol(bdy)|**2>) Where: volint(Bpol**2) = volume integral of Bpol**2 over the entire core plasma Pvol = volume of core plasma (i.e. mag. axis to boundary). mu0 = 1.25664D-6 Ip = plasma current Lpol = poloidal path length of plasma boundary RLI_3 = major radius defined as (Rmin+Rmax)/2 of the boundary and <|Bpol(bdy)|**2> = (Philim/pi*q(bdy))**2*<|grad(x)|**2/R**2> @ bdy with Philim = total enclosed toroidal flux, pi = 3.14159..., q is the rotational transform, the usual q profile, and the metric quantity GX2R2I = <|grad(x)|**2/R**2> is evaluated as a differential flux surface volume average in the usual way: int(d(Lpol)*(R/|grad(x)|)*|grad(x)|**2/R**2) <|grad(x)|**2/R**2> = ------------------------------------------- int(d(Lpol)*(R/|grad(x)|) i.e. the ratio of two loop integrals with differential volume weighting, evaluated numerically from the equilibrium flux surface geometry; x is the normalized sqrt(toroidal flux) radial coordinate (flux surface label with x=0 the magnetic axis, x=1 the core plasma boundary). The non-circular Li/2 and Beta(pol) outputs of TRANSP use the LI_VDIFF normalization. To convert to LI_1 normalization, apply the factor (LI_1/LI_VDIFF) to the quantity in question. To convert to LI_3 normalization, apply the factor (LI_3/LI_VDIFF).