A set of experimentally-determined dimensionless parameters is proposed for
characterizing the regime of divertor operation. The objective is to be able to
compare as unambiguously as possible the operation of different divertors and
to understand what physical similarities and differences they represent.
Examples from Alcator C-Mod are given.Comment: Plain Tex (8 pages) plus 5 postscipt figure

Hutchinson, I. H.

LaBombard, B.

Lipschultz, B.

English

arXiv

A set of experimentally-determined dimensionless parameters is proposed for characterizing the regime of divertor operation. The objective is to be able to compare as unambiguously as possible the operation of different divertors and to understand what physical similarities and differences they represent. Examples from Alcator C- Mod are given. 4 refs., 5 figs,. 3 tabs

Hutchinson, I.H.

UNT (University of North Texas) Digital Library

PFC/RR-95-12 DOE/ET-51013-316 Experimental Divertor Similarity Database Parameters I.H. Hutchinson, B. LaBombard, B. Lipschultz December 1995 DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or respnsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. This work was supported by the U. S. Department of Energy Contract No. DSACOZ 78ET51013. Reproduction, translation, publication, use and disposal, in whole or in part by or for the United States government is permitted. ~~~~~0~ OF THIS DOCUMENT IS UNLIMIED Experimental Divertor Similarity Database Parameters I.H. Hut chinson, B. LaBombard and B. Lipschult z Plasma Fusion Center, MIT, Cambridge MA, USA. Abstract A set of experiment ally-determined dimensionless parameters is proposed for characterizing the regime of divertor operation. The objective is to be able to compare as unambiguously as possible the operation of different divertors and to understand what physical similarities and differences they represent. Examples from Alcator C-Mod are given. Background Recent theory has shown that exactly scaled similarity experiments on tokamak divertors, where atomic physics is important in addition to plasma physics processes, are not feasible because there are no free parameters. However, similarity comparisons can be obtained in principle if either some of the plasma parameters [l], or some of the geometric parameters [Z], or both, are unimportant. Moreover, such comparisons can be made practical if only approximate similarity is required. Approaches to obtaining similarity with ITER divertor regimes have been explored under various assumptions about divertor transport scaling. There are now several moderately similar divertor tokamaks, with increasing quantities of divertor data, that span a range of dimensionless regimes. In addition to the theoret- ical approaches to identifying similarity scaling, it seems most natural now to examine the experimental results to discover what the values of relevant dimensionless parameters actually are, and therefore to try to compare from experiment to experiment so as to dis- cover the underlying physics that is similar or different in them. To do this requires a set of dimensionless parameters to be indentified and, as far as possible, measured on each experiment. With these stored in appropriate databases, it ought to be possible quickly to identify dimensionless regimes. Alternatively, the parameters can be evaluated from a less processed database of raw data once the form of the parameters are agreed, provided that the database contains the relevant raw data. The purpose of this note is to propose parameters that determine the regimes of the relevant physics, and are experimentally measurable. Since judgement must be exercised in the choice of the exact parameters to include in the set, these choices should not be regarded either as definitive or as exhaustive. However, it is hoped that the set will form the basis for discussion, initial comparisons, and further refinement. 1 The Formal Parameters As has been discussed elsewhere [1,2] the dimensionless parameters deriving from the intensive variables (i.e. density, temperature, magnetic field, etc.) of the plasma can be considered to be p*: The ratio of Larmor radius to characteristic size. v*:  The collisionality - ratio of size to mean-free-path. A&: The ratio of Debye length to size. p: The ratio of plasma kinetic to magnetic pressure. T*: The ratio of plasma temperature to characteristic atomic energy. fj: The ion (or neutral) density of a species, j, as a fraction of electron density. Of these, it is commonly supposed that XY, is unimportant. The parameters f j  are un- doubtedly important but often not explicitly considered. In addition to these parameters there is a host of other dimensionless parameters that can be considered to be geometric. These include 0 Relative shapes of the plasma and divertor cross-section, including such quantities as 0 Ratios of poloidal to toroidal field, and hence field-line angle, safety factor, and so on. Moreover, geometric considerations are essential for the intensive parameters too, because their values depend upon the choice of their exact definition and where they are measured. Therefore it is important to be explicit about how the intensive parameters are to be defined. The parameters listed above are sufficient to define the similarity of equivalent systems. If two systems are found with all these parameters the same, then all relevant parameters are equal for the two systems. However, in comparing systems which are not exactly similar, there are other dimensionless parameters that describe important processes in the divertor. It may be of interest to document their values explicitly even if they are in principle determined by the parameters above. We here propose one such vital parameter: the ratio of the charge-exchange mean-free-path to the SOL thickness. The following proposals for specific choices of parameters’ definitions are based on judgement about what is important and what it is generally measured. They are therefore open to refinement. However, it is hoped that some degree of agreement on parameter definitions can be obtained along these lines. divertor flux expansion factors and relative depth. 2 The Specific Parameters There are, roughly speaking, three poloidal positions in the SOL that may be considered characteristic. They are, (1) the main chamber, (2) the x-point, (3) the target. By the “main chamber”, we mean the part of the SOL that is adjacent to the core plasma and sufliciently far from the x-point as not to be significantly influenced by it. The x-point is the transition between regions of the SOL adjacent to the main plasma and to the private flux region. This point is representative of the upstream divertor. The target is where the field lines intersect the solid surface. It is proposed that intensive dimensionless parameters be defined at each of the three positions. Generally, in present experiments there are measurements at the target and in the main chamber but perhaps not at the x-point, although efforts are currently devoted to obtaining this data. The radial scale-length with which plasma characteristic lengths are to be compared is the width of the scrape-off layer, A. It is proposed that this be defined as half the pressure e-folding length. This e-folding length is the distance from the separatrix to where the plasma electron pressure (nT) is exp(-1) of its value at the separatrix, or, equivalently for an exponential profile, the ratio of the radial integral of the pressure to its separatrix value. The practical advantage of using the pressure is that, except for detached divertors, the pressure is approximately constant along the flux surface so the A can be evaluated either from main-chamber or from target measurements. (A potential disadvantage is that we need both T and n to be measured.) Half the exponential width is used so as to be typical of the near-separatrix region. Note, of course, that A varies with poloidal position. The flux surface at distance A is proposed also as the radial position at which the plasma parameters such as T, and ne are to be evaluated. Their values there will be slightly smaller (by about 20%) than the separatrix values in the main chamber, which are most often quoted. However, the advantage of using the values at A is that they are more representative of the whole SOL. It is found that the divertor target values exactly on the separatrix are very often strongly influenced by the private flux region and by neutrals, rendering them even less typical of the divertor fan as a whole. For the gyrosize parameter, p*, it is proposed that this be defined for singly charged ions as I I I This uses the ion Larmor radius at the sound speed, ignoring the ion temperature. Cor- rections for finite Ti can be incorporated as an auxiliary parameter Ti/Te. It also uses the toroidal field whose variation as a function of position is weak, so that the p* variation from place to plase is caused primarily by temperature differences. Physically, the poloidal Larmor radius may be key. It then requires the extra geometric parameter of the field-line 3 angle, Bp/Bt. The ratio of the local poloidal Larmor radius to the local scrape-off length is almost independent of poloidal position (because each has a Bt/Bp factor and they cancel) and is equal to p*Bt/Bp using midplane values. The Coulomb collisionality of greatest interest is that for parallel motions. It is pro- posed that the parallel connection length, L, for the magnetic field be documented for two regions: the main chamber and the divertor. We denote by L ,  the complete connection length from target to target and by L, the connection length fi-om x-point to target. (The inner and outer divertor connection lengths may be different in single-null configurations, and there are two different total connection lengths for double-null configurations.) The connection length has a logarithmic singularity at the separatrix arising from the x-point region. Connection lengths are therefore measured on the flux surface that is a distance A outside the separatrix, where the plasma parameters are measured. It is proposed that the mean-free-path for Coulomb collisions be taken as that for electron-ion momentum transfer, sometimes called “slowing down”. Using the plasma formulary [3] expression for thermal electrons, ve = 2.9 x 10-6(n/cm-3) In A(Te/eV)-3/2 and the characteristic speed, (Te/me)1/2 the electron mean-free-path is taking the coulomb logarithm, In A, to be 10. We then define for the different connection lengths. It should be noted that the ion-ion mean-free-path is essentially the same as the elec- tron, at the same temperature. The ion-electron mean-free-path is (mi/m,)1/2 longer. It should also be noted that the mean free path for epithermal electrons (v M 3.7ut) respon- sible for heat transfer is much longer than the thermal value, by a factor approximately 17 accounting for both electron and ion collisions and using the fast electron expressions. This latter collisionality determines whether the SOL is in the sheath- or conduction-limited regimes. For the dimensionless temperature, relative to characteristic atomic physics energy scale, there is no obvious atomic energy that ought to be prefered; therefore the tempera- ture in some convenient units is the natural choice of definition. Electron volts seems the natural choice. More importantly, the electron and ion temperatures may well be different. For the atomic processes the electron temperature is undoubtedly more important. The ion temperature is not completely insignificant, expecially for such processes as charge exchange, but its importance as a dimensionless parameter relative to atomic energies is much less. 4 The key geometric parameters are associated with the plasma shape and the field line angle. From the point of view of the scrape-off layer, the significant shape parameters are presumably (1) the ratio of the connection length in the divertor to the connection length in the main chamber and (2) the ratio of the flux surface spacing in the divertor to the spacing in the main chamber (i.e. the flux expansion factor), which we will choose to define at a distance A outside the separatrix so as to be well defined at the x-point poloidal position. These together also determine the ratio of the divertor poloidal depth to the plasma radius. Moreover, the flux expansion factor is equal to the ratio of the field line angles in the divertor and main chamber. However, one value of field line angle is still required to define the field geometry, it can be taken simply as Bp/Bt at the midplane. A parameter that may also be important for the details of recycling and related pro- cesses at the target is the angle in the poloidal plane of intersection between the separatrix and the divertor target. We will denote this c y p .  A “horizontal” target plate generally has ap < ~ / 2 ,  whiIe a “vertical” plate has ap > 7r/2. The total three-dimensional angle of intersection of the field-lie and the target (at say) is then given by sinat = sinap/(l + B:/Bi)1 /2 ,  evaluated at the target. An auxiliary parameter of great importance is the collisionality of neutral flow across the divertor. For symmetric charge-exchange of hydrogen, the reaction rate is <av> M 10-14T:-33 m-’s-l for temperatures of interest. Therefore, the mean-free-path can be taken as lcz = (2&/m)1/2/92<au> m, where & is the neutral energy. We can consider specified energy (e.g. cold or Franck-Condon) neutrals, or else hot thermal neutrals with E = Ti/2. We define representative cases as the cold expression for 3 eV atoms: and the hot expression for thermalized atoms: Electron excitation and ionization processes are also of great importance. However, the plethora of different rates makes the task of defining explicit excitation lengths overwhelm- ing. In order of magnitude, the ionization rate above about 10 eV temperature is about a factor of two less than the charge-exchange rate. So that the ionization length is similar to the charge-exchange length. However, as the electron temperature falls below 10 eV, ionization rapidly becomes negligible. We will now present expressions for the dimensional and dimensionless parameters in table form. We use slightly abbreviated notation for the dimensional parameters as noted in Table 1. Also p denotes the bulk ion mass in units of the proton mass. 5 Table 1. Dimensional Parameters. Parameter Units Definition 7x20 1020m-3 Electron density at A-surface T eV Electron temperature at A-surface Bt T Toroidal magnetic field L m  Parallel connection length A m  Half radial midplane distance to l /e  pressure point. Qualified by position subscripts: m main chamber, x x-point, and t target. Table 2. Dimensionless parameters in practical units (eV, m, T). Parameter Formula Description Relative parallel length of divertor. Flw expansion factor at target. Flux expansion factor at x-point. Field line angle in main chamber (midplane). Poloidal angle of intersection with target. p* v* 6.9 x io3* Parallel Coulomb Collisionality. p 8.0 x 10-~v SOL toroidal beta. T Temper at ure f j  njlne Relative species density. 1.02 x 10- 4 (CcT)"2 Inverse of gyrowidth of SOL. Ti / T e  Relative ion temperature. A/lcx,cold 42A~zzo(Ti /p)~-~~ Transverse cold neutral collisionality. A/lcx$ot 71An20(Ti/p)-~'~~ Transverse thermalized neutral collisionality. Non-geometric quantities qualified by position subscripts: m main chamber, x x-point, and t target. A spreadsheet [4] has been set up whereby the dimensionless parameters are calculated once the dimensional parameters of Table 1 and additional independent parameters (mostly geometric) of Table 2 are input. An illustrative example of this spreadsheet is given in Table 3. 6 Alcator C-Mod Examples The parameters defined above have been included in the Alcator C-Mod edge database and are calculated automatically, based primarily on probe data (embedded and fast- scanning probes). We illustrate the range of parameters on C-Mod by several figures based on operation during March through April 1995. Fig 1 shows a plot of poloidal field against separatrix density; the current grouping shows that the field depends primarily on plasma current, as expected. (This data is all at toroidal field of 5.3T.) Fig 2 plots the dimensionless parameters p* versus v*. The value of v* ranges from about 10 to over 400, corresponding to the observed range of regimes from sheath-limited to conduction-limited. Fig 3 plots p* versus A. The betas are low; the widths a few millimeters. All these data are main-chamber parameters (denoted ‘mid’ in the plots). In Fig 4 we show the total connection length (target to target) as a function of the ratio of A to the poloidal Larmor radius (i.e. (BP/Bt ) l /p* ) .  Connection length varies primarily with current. The SOL width is typically only a few times the poloidal gyrolength, thus emphasizing the importance of cross-field drifts, as has been experimentally observed. Fig 5 illustrates the values of the ‘cold’ neutral collisionality at the outer divertor (x-axis) and the main chamber (y-axis). The neutral collisionality is generally less than about 0.4 in the main chamber, implying that a significant fraction of neutrals can cross the SOL. At the divertor, however, the collisionality is much higher (particularly at high densities) and the divertor leg is therefore ‘thick’ to neutrals. These illustrative examples document the parameter ranges spanned by this set of experiments. It is intended that inter-machine comparisons be carried out using the defi- nitions described here. References [l] K.Lackner, Comments on Plasma Physics and Controlled Fusion, 15, 359 (1994). [2] 1.H.Hutchinson and G.C.Vlases, to be published in Nuclear Fusion. [3] D.L.Book, Plasma Fo~mulary, NRL publication 0084-4040 (1986). [4] This can be obtained in by anonymous FTP from //pfc.mit.edu/hutch/divsim.zip Figure Captions Figure 1 Plot of similarity database parameters (SIM-) poloidal field versus separatrix density (at midplane). Figure 2 Plot of p* versus v* (at midplane). Figure 3 Plot of p* versus A. Figure 4 Plot of total connection length versus Alp, (at midplane). Figure 5 Plot of cold neutral transverse collisionality (A/tcz,cold) at the midplane versus at the outer divertor. 7 Table 3. Spreadsheet f o r  Calculation of Experimental Divertor Similari ty Parameters Example Parameter Units Value B-t  T * 4.00E+00 L-m m * 2.20E+Ol L,x/L-m * 1.6OE-01 B-pt/B-pm * 3.7OE-01 B,px/B-pm * 1.40E-01 B-pdB-tm * 1.7OE-01 alpha-p * 5.00E-01 m- i/m,prot on * 2.OOE+OO Main chamber 11-20 T T,i/T,e Delta rho* nu* beta D/l-cold D/l,hot D/rho,p X-point n-20 T T,i/T,e Delta-x rho* nu* beta D/l,cold D/l,hot D/ rho -p Target n-20 T T,i/T,e Delta-t rho* nu* beta D/l,cold D/l,hot D /rho -p alpha-t 10-20 * 8.4OE-01 eV * 6.OOE+Ol * l.OOE+OO m * 1.6OE-03 1 .78E-0 1 3.543+01 2.52E-04 2.18E-01 4.76E-02 9.55E-01 10-20 * 1.4OE+OO eV * 4.00E+Ol * 1.00E+00 m 1.1%-02 2.03E-02 2.13E+Ol 2.8OE-04 2.27E+00 6.07E-01 8.35E+00 10-20 * 2.44E+OO eV * 5.20E+00 * l.OOE+OO m 4.323-03 1.943-02 2.19E+03 6.3%-05 7.64E-01 5.66E-01 8*77E+OO 3.OlE-02 Denotes input value * : 8  Figure 1. Plot of similarity database parameters (SIM-) poloidal field ver- sus separatrix density (at midplane). -1 c SI MNe_Dal ta: 0.8 1 (-1 LOCATION = MID 0.0 0.2 0.4 0.6 0.8 1 .o SIM-Ne-Delta ( 1  O2' m-3) 1.2 1.4 0 --. 0 0 crc 0 0 P 0 0 0 0 0 0 u1 0 SIMRhoStar  () 0 0 0 UI 0 0 iL 0 v, f CY ? 0 01 iQ i 0 v, U a + m I z T 7 E I II E? I n II U U  0 F 00 a 11 !2 iir 0 I ,,.>’.? .. . .: .., : 0 I I I I I I I I I  I I I I I I I I I  I I I I I I I I I  I I I I I I I I I  I I I I I I I I I  I I I I I I I I  0 m 0 m 0 d- 0 Fr) (w) 3 T W I S  0 c\l 0 0 ? CD 0 > 0 I -n Q I I I I I I I I I  I 1 1 1 1 1 1 I I  I I I I I I I I I  I I I I I I I I I  I I I I I I I I  - - - - - - 0 -  1- I u -  a -  a -  I x -  + -0 0 Ln 0 0 7 0 0 0 0 0 7 00 /1 W U c 0 13 

Experimental divertor similarity database parameters

Bombard, B.

DSpace@MIT

DOE/ET-51013-316Experimental DivertorSimilarity Database ParametersI.H. Hutchinson, B. LaBombard, B. LipschultzDecember 1995This work was supported by the U. S. Department of Energy Contract No. DE-AC02-78ET51013. Reproduction, translation, publication, use and disposal, in whole or in partby or for the United States government is permitted.PFC/RR-95-12Experimental Divertor Similarity Database ParametersI.H.Hutchinson, B.LaBombard and B.LipschultzPlasma Fusion Center, MIT, Cambridge MA, USA.AbstractA set of experimentally-determined dimensionless parameters is proposed for characterizingthe regime of divertor operation. The objective is to be able to compare as unambiguouslyas possible the operation of different divertors and to understand what physical similaritiesand differences they represent. Examples from Alcator C-Mod are given.BackgroundRecent theory has shown that exactly scaled similarity experiments on tokamak divertors,where atomic physics is important in addition to plasma physics processes, are not feasiblebecause there are no free parameters. However, similarity comparisons can be obtained inprinciple if either some of the plasma parameters [1], or some of the geometric parameters[2], or both, are unimportant. Moreover, such comparisons can be made practical if onlyapproximate similarity is required. Approaches to obtaining similarity with ITER divertorregimes have been explored under various assumptions about divertor transport scaling.There are now several moderately similar divertor tokamaks, with increasing quantitiesof divertor data, that span a range of dimensionless regimes. In addition to the theoret-ical approaches to identifying similarity scaling, it seems most natural now to examinethe experimental results to discover what the values of relevant dimensionless parametersactually are, and therefore to try to compare from experiment to experiment so as to dis-cover the underlying physics that is similar or different in them. To do this requires a setof dimensionless parameters to be indentified and, as far as possible, measured on eachexperiment. With these stored in appropriate databases, it ought to be possible quicklyto identify dimensionless regimes. Alternatively, the parameters can be evaluated from aless processed database of raw data once the form of the parameters are agreed, providedthat the database contains the relevant raw data.The purpose of this note is to propose parameters that determine the regimes of therelevant physics, and are experimentally measurable. Since judgement must be exercisedin the choice of the exact parameters to include in the set, these choices should not beregarded either as definitive or as exhaustive. However, it is hoped that the set will formthe basis for discussion, initial comparisons, and further refinement.1The Formal ParametersAs has been discussed elsewhere [1,2] the dimensionless parameters deriving from theintensive variables (i.e. density, temperature, magnetic field, etc.) of the plasma can beconsidered to bep*: The ratio of Larmor radius to characteristic size.V*: The collisionality - ratio of size to mean-free-path.A*: The ratio of Debye length to size.P: The ratio of plasma kinetic to magnetic pressure.T*: The ratio of plasma temperature to characteristic atomic energy.fj: The ion (or neutral) density of a species, j, as a fraction of electron density.Of these, it is commonly supposed that A* is unimportant. The parameters fj are un-doubtedly important but often not explicitly considered.In addition to these parameters there is a host of other dimensionless parameters thatcan be considered to be geometric. These include" Relative shapes of the plasma and divertor cross-section, including such quantities asdivertor flux expansion factors and relative depth." Ratios of poloidal to toroidal field, and hence field-line angle, safety factor, and so on.Moreover, geometric considerations are essential for the intensive parameters too, becausetheir values depend upon the choice of their exact definition and where they are measured.Therefore it is important to be explicit about how the intensive parameters are to bedefined.The parameters listed above are sufficient to define the similarity of equivalent systems.If two systems are found with all these parameters the same, then all relevant parametersare equal for the two systems. However, in comparing systems which are not exactlysimilar, there are other dimensionless parameters that describe important processes in thedivertor. It may be of interest to document their values explicitly even if they are inprinciple determined by the parameters above. We here propose one such vital parameter:the ratio of the charge-exchange mean-free-path to the SOL thickness.The following proposals for specific choices of parameters' definitions are based onjudgement about what is important and what it is generally measured. They are thereforeopen to refinement. However, it is hoped that some degree of agreement on parameterdefinitions can be obtained along these lines.2The Specific ParametersThere are, roughly speaking, three poloidal positions in the SOL that may be consideredcharacteristic. They are, (1) the main chamber, (2) the x-point, (3) the target. By the"main chamber", we mean the part of the SOL that is adjacent to the core plasma andsufficiently far from the x-point as not to be significantly influenced by it. The x-point isthe transition between regions of the SOL adjacent to the main plasma and to the privateflux region. This point is representative of the upstream divertor. The target is where thefield lines intersect the solid surface.It is proposed that intensive dimensionless parameters be defined at each of the threepositions. Generally, in present experiments there are measurements at the target and inthe main chamber but perhaps not at the x-point, although efforts are currently devotedto obtaining this data.The radial scale-length with which plasma characteristic lengths are to be compared isthe width of the scrape-off layer, A. It is proposed that this be defined as half the pressuree-folding length. This e-folding length is the distance from the separatrix to where theplasma electron pressure (nT) is exp(-1) of its value at the separatrix, or, equivalentlyfor an exponential profile, the ratio of the radial integral of the pressure to its separatrixvalue. The practical advantage of using the pressure is that, except for detached divertors,the pressure is approximately constant along the flux surface so the A can be evaluatedeither from main-chamber or from target measurements. (A potential disadvantage is thatwe need both T and n to be measured.) Half the exponential width is used so as to betypical of the near-separatrix region. Note, of course, that A varies with poloidal position.The flux surface at distance A is proposed also as the radial position at which theplasma parameters such as Te and ne are to be evaluated. Their values there will beslightly smaller (by about 20%) than the separatrix values in the main chamber, which aremost often quoted. However, the advantage of using the values at A is that they are morerepresentative of the whole SOL. It is found that the divertor target values exactly on theseparatrix are very often strongly influenced by the private flux region and by neutrals,rendering them even less typical of the divertor fan as a whole.For the gyrosize parameter, p*, it is proposed that this be defined for singly chargedions as(m2 T.) /eBtThis uses the ion Larmor radius at the sound speed, ignoring the ion temperature. Cor-rections for finite Ti can be incorporated as an auxiliary parameter Ti/Te. It also uses thetoroidal field whose variation as a function of position is weak, so that the p* variationfrom place to plase is caused primarily by temperature differences. Physically, the poloidalLarmor radius may be key. It then requires the extra geometric parameter of the field-line3angle, B,/Bt. The ratio of the local poloidal Larmor radius to the local scrape-off length isalmost independent of poloidal position (because each has a Bt/Bp factor and they cancel)and is equal to p*Bt/Bp using midplane values.The Coulomb collisionality of greatest interest is that for parallel motions. It is pro-posed that the parallel connection length, L, for the magnetic field be documented for tworegions: the main chamber and the divertor. We denote by Lm the complete connectionlength from target to target and by L, the connection length from x-point to target. (Theinner and outer divertor connection lengths may be different in single-null configurations,and there are two different total connection lengths for double-null configurations.) Theconnection length has a logarithmic singularity at the separatrix arising from the x-pointregion. Connection lengths are therefore measured on the flux surface that is a distanceA outside the separatrix, where the plasma parameters are measured. It is proposed thatthe mean-free-path for Coulomb collisions be taken as that for electron-ion momentumtransfer, sometimes called "slowing down". Using the plasma formulary [3] expression forthermal electrons, Ve = 2.9 x 10-6 (n/cm- 3) ln A(Te/eV)- 3/2 and the characteristic speed,(Te/me)1/2 the electron mean-free-path isee = 1.45 x 10-4 m(T/eV)2(n/102 0 m-3 ) Mtaking the coulomb logarithm, ln A, to be 10. We then defineV* = L/X ,for the different connection lengths.It should be noted that the ion-ion mean-free-path is essentially the same as the elec-tron, at the same temperature. The ion-electron mean-free-path is (mi/me)1/2 longer. Itshould also be noted that the mean free path for epithermal electrons (v P 3.7vt) respon-sible for heat transfer is much longer than the thermal value, by a factor approximately 17accounting for both electron and ion collisions and using the fast electron expressions. Thislatter collisionality determines whether the SOL is in the sheath- or conduction-limitedregimes.For the dimensionless temperature, relative to characteristic atomic physics energyscale, there is no obvious atomic energy that ought to be prefered; therefore the tempera-ture in some convenient units is the natural choice of definition. Electron volts seems thenatural choice. More importantly, the electron and ion temperatures may well be different.For the atomic processes the electron temperature is undoubtedly more important. Theion temperature is not completely insignificant, expecially for such processes as chargeexchange, but its importance as a dimensionless parameter relative to atomic energies ismuch less.4The key geometric parameters are associated with the plasma shape and the field lineangle. From the point of view of the scrape-off layer, the significant shape parametersare presumably (1) the ratio of the connection length in the divertor to the connectionlength in the main chamber and (2) the ratio of the flux surface spacing in the divertorto the spacing in the main chamber (i.e. the flux expansion factor), which we will chooseto define at a distance A outside the separatrix so as to be well defined at the x-pointpoloidal position. These together also determine the ratio of the divertor poloidal depthto the plasma radius. Moreover, the flux expansion factor is equal to the ratio of the fieldline angles in the divertor and main chamber. However, one value of field line angle is stillrequired to define the field geometry, it can be taken simply as Bp/Bt at the midplane.A parameter that may also be important for the details of recycling and related pro-cesses at the target is the angle in the poloidal plane of intersection between the separatrixand the divertor target. We will denote this a,. A "horizontal" target plate generallyhas a, < r/2, while a "vertical" plate has a, > 7r/2. The total three-dimensionalangle of intersection of the field-line and the target (&t say) is then given by sinat =sin ap/(1 + B2/B2)1/ 2 , evaluated at the target.An auxiliary parameter of great importance is the collisionality of neutral flow acrossthe divertor. For symmetric charge-exchange of hydrogen, the reaction rate is <uv> ;10- 4 T. 33 m- 3s-' for temperatures of interest. Therefore, the mean-free-path can betaken as = (2E/m)1/2/n<av> m, where F is the neutral energy. We can considerspecified energy (e.g. cold or Franck-Condon) neutrals, or else hot thermal neutrals withE = T/2. We define representative cases as the cold expression for 3 eV atoms:cx,cold ~ 2.4 x 10-2 [(T-.eV)]- 3 3 (n/10 2 0 m 3 )- mand the hot expression for thermalized atoms:4cx,hot ; 1.4 x 10-2[(T/eV)/p]0 - (ni/1020 m-3)-l m.Electron excitation and ionization processes are also of great importance. However, theplethora of different rates makes the task of defining explicit excitation lengths overwhelm-ing. In order of magnitude, the ionization rate above about 10 eV temperature is about afactor of two less than the charge-exchange rate. So that the ionization length is similarto the charge-exchange length. However, as the electron temperature falls below 10 eV,ionization rapidly becomes negligible.We will now present expressions for the dimensional and dimensionless parameters intable form. We use slightly abbreviated notation for the dimensional parameters as notedin Table 1. Also p denotes the bulk ion mass in units of the proton mass.5Table 1. Dimensional Parameters.Parameter Units Definitionn 2 0  10 2 0 m-3  Electron density at A-surfaceT eV Electron temperature at A-surfaceBt T Toroidal magnetic fieldL m Parallel connection lengthA m Half radial midplane distance to 1/e pressure point.Qualified by position subscripts: m main chamber, x x-point, and t target.Table 2. Dimensionless parameters in practical units (eV, m, T).Parameter Formula DescriptionL,/Lm Relative parallel length of divertor.Bpt/Bpm Flux expansion factor at target.Bp,/Bpm Flux expansion factor at x-point.Bpm/Btm Field line angle in main chamber (midplane)..'p Poloidal angle of intersection with target.p* 1.02 x 10-4 (pT)1 2  Inverse of gyrowidth of SOL.X Av* 6.9 x 103 - Parallel Coulomb Collisionality.# 8.0 x 10-5 n SOL toroidal beta.T TemperatureTi/Te Relative ion temperature.fi n /ne Relative species density.A/ecx,cold 42An 20(T /p)0.33 Transverse cold neutral collisionality.A/tcx,hot 71An 20(Ti/1)-0-17 Transverse thermalized neutral collisionality.Non-geometric quantities qualified by position subscripts: m main chamber, x x-point,and t target.A spreadsheet [4] has been set up whereby the dimensionless parameters are calculatedonce the dimensional parameters of Table 1 and additional independent parameters (mostlygeometric) of Table 2 are input. An illustrative example of this spreadsheet is given inTable 3.6Alcator C-Mod ExamplesThe parameters defined above have been included in the Alcator C-Mod edge databaseand are calculated automatically, based primarily on probe data (embedded and fast-scanning probes). We illustrate the range of parameters on C-Mod by several figuresbased on operation during March through April 1995. Fig 1 shows a plot of poloidal fieldagainst separatrix density; the current grouping shows that the field depends primarily onplasma current, as expected. (This data is all at toroidal field of 5.3T.) Fig 2 plots thedimensionless parameters p* versus v*. The value of v* ranges from about 10 to over 400,corresponding to the observed range of regimes from sheath-limited to conduction-limited.Fig 3 plots .0* versus A. The betas are low; the widths a few millimeters. All these dataare main-chamber parameters (denoted 'mid' in the plots).In Fig 4 we show the total connection length (target to target) as a function of the ratioof A to the poloidal Larmor radius (i.e. (B,/Bt)1/p*). Connection length varies primarilywith current. The SOL width is typically only a few times the poloidal gyrolength, thusemphasizing the importance of cross-field drifts, as has been experimentally observed. Fig5 illustrates the values of the 'cold' neutral collisionality at the outer divertor (x-axis) andthe main chamber (y-axis). The neutral collisionality is generally less than about 0.4 inthe main chamber, implying that a significant fraction of neutrals can cross the SOL. Atthe divertor, however, the collisionality is much higher (particularly at high densities) andthe divertor leg is therefore 'thick' to neutrals.These illustrative examples document the parameter ranges spanned by this set ofexperiments. It is intended that inter-machine comparisons be carried out using the defi-nitions described here.References[1] K.Lackner, Comments on Plasma Physics and Controlled Fusion, 15, 359 (1994).[2] I.H.Hutchinson and G.C.Vlases, to be published in Nuclear Fusion.[3] D.L.Book, Plasma Formulary, NRL publication 0084-4040 (1986).[4] This can be obtained in by anonymous FTP from //pfc.mit.edu/hutch/divsim.zipFigure CaptionsFigure 1 Plot of similarity database parameters (SIM.) poloidal field versus separatrixdensity (at midplane).Figure 2 Plot of p* versus v* (at midplane).Figure 3 Plot of #* versus A.Figure 4 Plot of total connection length versus A/p, (at midplane).Figure 5 Plot of cold neutral transverse collisionality (A/gc,cold) at the midplane versusat the outer divertor.7Table 3. Spreadsheet for Calculation ofExperimental Divertor Similarity ParametersExampleParameter Units ValueB.t T * 4.OOE+00L-m m * 2.20E+01L-x/L-m * 1.60E-01B.pt/B-pm * 3.70E-01B-px/B.pm * 1.40E-01B.pm/B-tm * 1.70E-01alpha-p * 5.OOE-01m-i/m.proton * 2.OOE+00Main chambern-20 102O *0T eV *T-i/T-e*Delta m *rho*nu*betaD/LcoldD/LhotD/rho.pX-pointn-20 10^2 *T eV *Ti/Te *Delta-x mrho*nu*betaD/1.coldD/LhotD/rho.pTargetn.20 10^2 *T eV *T-i/T-e*Delta-t mrho*nu*betaD/1-coldD/1.hotD/rho.palpha-tDenotes input value *8.40E-O16.OOE+O11.OOE+001.60E-031.78E-013.54E+012.52E-042.18E-014.76E-029.55E-011. 40E+004.OOE+011.OOE+001.14E-022.03E-022.13E+012.80E-042.27E+006.07E-018.35E+002.44E+005.20E+001. OOE+004.32E-031.94E-022.19E+036.34E-057.64E-015.66E-018.77E+003.O1E-02: 8I f I I I I I0~G 2iiM2---IX0 0]I0)0I I I I ii II I I Id8-VISC~400(0D0(DIS~i)9K>'acnE0NoCD00,000 00;.0 $:4 4-1oI.*' Q)049 1o E0c00C) 0V)l 0-D-4-t 000it L( 0N N '- 0 Mo o o 0 6() .. 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Experimental Divertor Similarity Database Parameters

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