From 9cd8bcd468d7bfde947fffc7876970be42d02390 Mon Sep 17 00:00:00 2001 From: Stuart Muldrew Date: Thu, 18 Jul 2024 15:08:45 +0100 Subject: [PATCH 1/8] Updated iprofile options and removed gtscale --- process/physics.py | 61 ++++++++++++++++------------ source/fortran/input.f90 | 7 +--- source/fortran/physics_variables.f90 | 19 ++++----- 3 files changed, 45 insertions(+), 42 deletions(-) diff --git a/process/physics.py b/process/physics.py index ff9fb795a9..9385cf8ce1 100644 --- a/process/physics.py +++ b/process/physics.py @@ -1381,28 +1381,27 @@ def physics(self): ) # Calculate physics_variables.beta limit - if physics_variables.iprofile == 0: - if physics_variables.gtscale == 1: - # Original scaling law - physics_variables.dnbeta = 2.7e0 * ( - 1.0e0 + 5.0e0 * physics_variables.eps**3.5e0 - ) - - if physics_variables.gtscale == 2: - # See Issue #1439 - # physics_variables.dnbeta found from physics_variables.aspect ratio scaling on p32 of Menard: - # Menard, et al. "Fusion Nuclear Science Facilities - # and Pilot Plants Based on the Spherical Tokamak." - # Nucl. Fusion, 2016, 44. - physics_variables.dnbeta = ( - 3.12e0 + 3.5e0 * physics_variables.eps**1.7e0 - ) - else: + if physics_variables.iprofile == 1: # Relation between physics_variables.beta limit and plasma internal inductance # Hartmann and Zohm physics_variables.dnbeta = 4.0e0 * physics_variables.rli + if physics_variables.iprofile == 2: + # Original scaling law + physics_variables.dnbeta = 2.7e0 * ( + 1.0e0 + 5.0e0 * physics_variables.eps**3.5e0 + ) + + if physics_variables.iprofile == 3 or physics_variables.iprofile == 5: + # physics_variables.dnbeta found from physics_variables.aspect ratio scaling on p32 of Menard: + # Menard, et al. "Fusion Nuclear Science Facilities + # and Pilot Plants Based on the Spherical Tokamak." + # Nucl. Fusion, 2016, 44. + physics_variables.dnbeta = ( + 3.12e0 + 3.5e0 * physics_variables.eps**1.7e0 + ) + physics_variables.betalim = culblm( physics_variables.bt, physics_variables.dnbeta, @@ -2030,8 +2029,12 @@ def culcur( 8 = Sauter scaling (allowing negative triangularity) Issue #392 'Geometric formulas for system codes including the effect of negative triangularity' iprofile : input integer : switch for current profile consistency - 0 = use input alphaj, rli - 1 = make these consistent with q, q0 + 0 use input values for alphaj, rli, dnbeta + 1 make these consistent with input q, q_0 values (recommend `icurr=4` with this option) + 2 use input values for alphaj, rli. Scale dnbeta with aspect ratio (original scaling) + 3 use input values for alphaj, rli. Scale dnbeta with aspect ratio (Menard scaling) + 4 use input values for alphaj, dnbeta. Set rli from elongation (Menard scaling) + 5 use input value for alphaj. Set rli and dnbeta from Menard scaling kappa : input real : plasma elongation kappa95 : input real : plasma elongation at 95% surface p0 : input real : central plasma pressure (Pa) @@ -2151,12 +2154,20 @@ def culcur( icurr, plascur, q95, asp, eps, bt, kappa, triang, pperim, constants.rmu0 ) - # Ensure current profile consistency, if required - # This is as described in Hartmann and Zohm only if icurr = 4 as well... + if iprofile == 1: + # Ensure current profile consistency, if required + # This is as described in Hartmann and Zohm only if icurr = 4 as well... + + # Tokamaks 4th Edition, Wesson, page 116 alphaj = qstar / q0 - 1.0 - rli = np.log(1.65 + 0.89 * alphaj) # Tokamaks 4th Edition, Wesson, page 116 + rli = np.log(1.65 + 0.89 * alphaj) + + if iprofile == 4 or iprofile == 5: + # Spherical Tokamak relation for internal inductance + # Menard et al. (2016), Nuclear Fusion, 56, 106023 + rli = 3.4 - kappa return alphaj, rli, bp, qstar, plascur @@ -2494,15 +2505,15 @@ def outplas(self): po.osubhd(self.outfile, "Current and Field :") if stellarator_variables.istell == 0: - if physics_variables.iprofile == 0: + if physics_variables.iprofile == 1: po.ocmmnt( self.outfile, - "Consistency between q0,q,alphaj,rli,dnbeta is not enforced", + "Consistency between q0,q,alphaj,rli,dnbeta is enforced", ) else: po.ocmmnt( self.outfile, - "Consistency between q0,q,alphaj,rli,dnbeta is enforced", + "Consistency between q0,q,alphaj,rli,dnbeta is not enforced", ) po.oblnkl(self.outfile) diff --git a/source/fortran/input.f90 b/source/fortran/input.f90 index 0993fc8816..8ba201d3d2 100644 --- a/source/fortran/input.f90 +++ b/source/fortran/input.f90 @@ -309,7 +309,7 @@ subroutine parse_input_file(in_file,out_file,show_changes) fgwsep, rhopedn, tratio, q0, ishape, fne0, ignite, ftrit, & ifalphap, tauee_in, alphaj, alphat, icurr, q, ti, tesep, rli, triang, & itart, ralpne, iprofile, triang95, rad_fraction_sol, betbm0, protium, & - teped, fhe3, iwalld, gamma, falpha, fgwped, gtscale, tbeta, ibss, & + teped, fhe3, iwalld, gamma, falpha, fgwped, tbeta, ibss, & iradloss, te, alphan, rmajor, kappa, iinvqd, fkzohm, beamfus0, & tauratio, idensl, ieped, bt, iscrp, ipnlaws, betalim, betalim_lower, & idia, ips, m_s_limit, burnup_in @@ -612,9 +612,6 @@ subroutine parse_input_file(in_file,out_file,show_changes) case ('gamma') call parse_real_variable('gamma', gamma, 0.1D0, 1.0D0, & 'Ejima coefficient for resistive V-s formula') - case ('gtscale') - call parse_int_variable('gtscale', gtscale, 0, 2, & - 'Flag to scale beta coefficient with R/a') case ('hfact') call parse_real_variable('hfact', hfact, 0.01D0, 10.0D0, & 'Energy confinement time H factor') @@ -653,7 +650,7 @@ subroutine parse_input_file(in_file,out_file,show_changes) call parse_int_variable('ipedestal', ipedestal, 0, 1, & 'Switch for plasma profile type') case ('iprofile') - call parse_int_variable('iprofile', iprofile, 0, 1, & + call parse_int_variable('iprofile', iprofile, 0, 5, & 'Switch for current profile consistency') case ('ips') call parse_int_variable('ips', ips, 0, 1, & diff --git a/source/fortran/physics_variables.f90 b/source/fortran/physics_variables.f90 index 513dc094ad..3dce3bc7c3 100644 --- a/source/fortran/physics_variables.f90 +++ b/source/fortran/physics_variables.f90 @@ -27,7 +27,7 @@ module physics_variables !! average mass of all ions (amu) real(dp) :: alphaj - !! current profile index (calculated from q_0, q if `iprofile=1`) + !! current profile index (calculated from q_0 and q if `iprofile=1`) real(dp) :: alphan !! density profile index @@ -124,8 +124,7 @@ module physics_variables !! hot beam ion density from calculation (/m3) real(dp) :: dnbeta - !! Troyon-like coefficient for beta scaling calculated - !! as 4*rli if `iprofile=1` (see also gtscale option) + !! Troyon-like coefficient for beta scaling real(dp) :: dnelimt !! density limit (/m3) @@ -231,13 +230,6 @@ module physics_variables real(dp) :: gammaft !! ratio of (fast alpha + neutral beam beta) to thermal beta - integer :: gtscale - !! switch for a/R scaling of dnbeta (`iprofile=0` only): - !! - !! - =0 do not scale dnbeta with eps - !! - =1 scale dnbeta with eps, original scaling - !! - =2 scale dnbeta with eps, Menard scaling - real(dp), dimension(ipnlaws) :: hfac !! H factors for an ignited plasma for each energy confinement time scaling law @@ -379,8 +371,12 @@ module physics_variables integer :: iprofile !! switch for current profile consistency: !! - !! - =0 use input values for alphaj, rli, dnbeta (but see gtscale option) + !! - =0 use input values for alphaj, rli, dnbeta !! - =1 make these consistent with input q, q_0 values (recommend `icurr=4` with this option) + !! - =2 use input values for alphaj, rli. Scale dnbeta with aspect ratio (original scaling) + !! - =3 use input values for alphaj, rli. Scale dnbeta with aspect ratio (Menard scaling) + !! - =4 use input values for alphaj, dnbeta. Set rli from elongation (Menard scaling) + !! - =5 use input value for alphaj. Set rli and dnbeta from Menard scaling integer :: iradloss !! switch for radiation loss term usage in power balance (see User Guide): @@ -954,7 +950,6 @@ subroutine init_physics_variables fvsbrnni = 1.0D0 gamma = 0.4D0 gammaft = 0.0D0 - gtscale = 0 hfac = 0.0D0 hfact = 1.0D0 taumax = 10.0D0 From 1f61cf8da9d2defe9ee518e517010714b432e875 Mon Sep 17 00:00:00 2001 From: Stuart Muldrew Date: Thu, 18 Jul 2024 15:53:17 +0100 Subject: [PATCH 2/8] Updated documentation --- documentation/proc-pages/index.md | 4 +-- .../proc-pages/physics-models/plasma_beta.md | 26 ++++++++++------- .../physics-models/plasma_current.md | 28 +++++++++++++------ 3 files changed, 36 insertions(+), 22 deletions(-) diff --git a/documentation/proc-pages/index.md b/documentation/proc-pages/index.md index 2f8b8a2b01..7202d84789 100644 --- a/documentation/proc-pages/index.md +++ b/documentation/proc-pages/index.md @@ -7,7 +7,7 @@ class="logo" > -PROCESS is a systems code at [CCFE](https://ccfe.ukaea.uk/) that calculates in a +PROCESS is a systems code at [UKAEA](https://ccfe.ukaea.uk/) that calculates in a self-consistent manner the parameters of a fusion power plant with a specified performance, ensuring that its operating limits are not violated, and with the option to optimise to a given function of these parameters. @@ -72,8 +72,6 @@ to anyone using PROCESS outputs or models based on them. - [Stuart Muldrew](mailto:stuart.muldrew@ukaea.uk) - [Alex Pearce](mailto:alexander.pearce@ukaea.uk) - [Jonathan Maddock](mailto:jonathan.maddock@ukaea.uk) -- [Charles Griesel](mailto:charles.griesel@ukaea.uk) -- [Rhian Chapman](mailto:rhian.chapman@ukaea.uk) - [Timothy Nunn](mailto:timothy.nunn@ukaea.uk) - [Christopher Ashe](mailto:christopher.ashe@ukaea.uk) - [Georgina Graham](mailto:georgina.graham@ukaea.uk) diff --git a/documentation/proc-pages/physics-models/plasma_beta.md b/documentation/proc-pages/physics-models/plasma_beta.md index 75198cd648..e28f3946f6 100644 --- a/documentation/proc-pages/physics-models/plasma_beta.md +++ b/documentation/proc-pages/physics-models/plasma_beta.md @@ -20,19 +20,21 @@ By default, $\beta$ is defined with respect to the total equilibrium B-field [^2 | 2 | Apply the $\beta$ limit to only the thermal plus neutral beam contributions to beta | | 3 | Apply the $\beta$ limit to the total beta (including the contribution from fast ions), calculated using only the toroidal field | -### Scaling of beta $g$ coefficient +### Setting the beta $g$ coefficient -Switch `gtscale` determines how the beta $g$ coefficient `dnbeta` should -be calculated, using the inverse aspect ratio $\epsilon = a/R$. +Switch `iprofile` determines how the beta $g$ coefficient `dnbeta` should +be calculated. -| `gtscale` | Description | +| `iprofile` | Description | | :-: | - | -| 0 | `dnbeta` is an input. | -| 1 | $g=2.7(1+5\epsilon^{3.5})$ (which gives g = 3.0 for aspect ratio = 3) | -| 2 | $g=3.12+3.5\epsilon^{1.7}$ (based on Menard et al. "Fusion Nuclear Science Facilities and Pilot Plants Based on the Spherical Tokamak", Nucl. Fusion, 2016, 44) | +| 0 | `alphaj`, `rli` and `dnbeta` are inputs. | +| 1 | `alphaj`, `rli` and `dnbeta` are calulcated consistently. `dnbeta` calculated using $g=4l_i$ [^3]| +| 2 | `alphaj` and `rli` are inputs. `dnbeta` calculated using $g=2.7(1+5\epsilon^{3.5})$ (which gives g = 3.0 for aspect ratio = 3) | +| 3 | `alphaj` and `rli` are inputs. `dnbeta` calculated using $g=3.12+3.5\epsilon^{1.7}$ [^4]| +| 4 | `alphaj` and `dnbeta` are inputs. `rli` calculated from elongation [^4]| +| 5 | `alphaj` is an input inputs. `rli` calculated from elongation and `dnbeta` calculated using $g=3.12+3.5\epsilon^{1.7}$ [^4]| -!!! Note - `gtscale` is over-ridden if `iprofile` = 1. +Further details on the calculation of `alphaj` and `rli` is given in [Plasma Current](./plasma_current.md). ### Limiting $\epsilon\beta_p$ @@ -43,4 +45,8 @@ is be set using input parameter `epbetmax`. [^1]: N.A. Uckan and ITER Physics Group, 'ITER Physics Design Guidelines: 1989', -[^2]: D.J. Ward, 'PROCESS Fast Alpha Pressure', Work File Note F/PL/PJK/PROCESS/CODE/050 \ No newline at end of file +[^2]: D.J. Ward, 'PROCESS Fast Alpha Pressure', Work File Note F/PL/PJK/PROCESS/CODE/050 + +[^3]: Tokamaks 4th Edition, Wesson, page 116 + +[^4]: Menard et al. (2016), Nuclear Fusion, 56, 106023 \ No newline at end of file diff --git a/documentation/proc-pages/physics-models/plasma_current.md b/documentation/proc-pages/physics-models/plasma_current.md index 2d5345d718..2be3309869 100644 --- a/documentation/proc-pages/physics-models/plasma_current.md +++ b/documentation/proc-pages/physics-models/plasma_current.md @@ -1,6 +1,6 @@ # Plasma Current Scaling Laws -A number of plasma current scaling laws are available in PROCESS $[^1]. These are calculated in +A number of plasma current scaling laws are available in PROCESS [^1]. These are calculated in routine `culcur`, which is called by `physics`. The safety factor $q_{95}$ required to prevent disruptive MHD instabilities dictates the plasma current Ip: @@ -39,10 +39,20 @@ $$\begin{aligned} l_i = ln(1.65+0.89\alpha_J) \end{aligned}$$ -The beta $g$ coefficient `dnbeta` also scales with $l_i$, as described above. +$$\begin{aligned} +g = 4 l_i +\end{aligned}$$ It is recommended that current scaling law `icurr = 4` is used if `iprofile = 1`. -Switch `gtscale` is over-ridden if `iprofile = 1`. +This relation is only applicable to large aspect ratio tokamaks. + +For spherical tokamaks, the internal inductance can be set from the elongation using `iprofile = 4` or `iprofile = 5`: + +$$\begin{aligned} +l_i = 3.4 - \kappa_x +\end{aligned}$$ + +Further desciption of `iprofile` is given in [Beta Limit](./plasma_beta.md). ## Bootstrap, Diamagnetic and Pfirsch-Schlüter Current Scalings @@ -56,8 +66,8 @@ existence of pedestals, whereas the Sauter et al. scaling | :-: | - | | 1 | ITER scaling -- To use the ITER scaling method for the bootstrap current fraction. Set `bscfmax` to the maximum required bootstrap current fraction ($\leq 1$). This method is valid at high aspect ratio only. | 2 | General scaling -- To use a more general scaling method, set `bscfmax` to the maximum required bootstrap current fraction ($\leq 1$). -| 3 | Numerically fitted scaling [^7] -- To use a numerically fitted scaling method, valid for all aspect ratios, set `bscfmax` to the maximum required bootstrap current fraction ($\leq 1$). -| 4 | Sauter, Angioni and Lin-Liu scaling [^8] [^9] -- Set `bscfmax` to the maximum required bootstrap current fraction ($\leq 1$). +| 3 | Numerically fitted scaling [^8] -- To use a numerically fitted scaling method, valid for all aspect ratios, set `bscfmax` to the maximum required bootstrap current fraction ($\leq 1$). +| 4 | Sauter, Angioni and Lin-Liu scaling [^9] [^10] -- Set `bscfmax` to the maximum required bootstrap current fraction ($\leq 1$). !!! Note "Fixed Bootstrap Current" Direct input -- To input the bootstrap current fraction directly, set `bscfmax` @@ -101,7 +111,7 @@ Unpublished internal Oak Ridge document. Current Drive', ITER-TN-PH-8-4, 13--17 June 1988, Garching, FRG [^6]: Y. Sakamoto, 'Recent progress in vertical stability analysis in JA', Task meeting EU-JA #16, Fusion for Energy, Garching, 24--25 June 2014 - -[^7]: H.R. Wilson, Nuclear Fusion **32** (1992) 257 -[^8]: O. Sauter, C. Angioni and Y.R. Lin-Liu, Physics of Plasmas **6** (1999) 2834 -[^9]: O. Sauter, C. Angioni and Y.R. Lin-Liu, Physics of Plasmas **9** (2002) 5140 \ No newline at end of file +[^7] Menard et al. (2016), Nuclear Fusion, 56, 106023 +[^8]: H.R. Wilson, Nuclear Fusion **32** (1992) 257 +[^9]: O. Sauter, C. Angioni and Y.R. Lin-Liu, Physics of Plasmas **6** (1999) 2834 +[^10]: O. Sauter, C. Angioni and Y.R. Lin-Liu, Physics of Plasmas **9** (2002) 5140 \ No newline at end of file From ccfbc85a8f464074a7228017e66d536b0326e597 Mon Sep 17 00:00:00 2001 From: Stuart Muldrew Date: Thu, 18 Jul 2024 16:07:24 +0100 Subject: [PATCH 3/8] Updated documentation --- documentation/proc-pages/physics-models/plasma_beta.md | 8 ++++---- documentation/proc-pages/physics-models/plasma_current.md | 4 ++-- 2 files changed, 6 insertions(+), 6 deletions(-) diff --git a/documentation/proc-pages/physics-models/plasma_beta.md b/documentation/proc-pages/physics-models/plasma_beta.md index e28f3946f6..048d17f75f 100644 --- a/documentation/proc-pages/physics-models/plasma_beta.md +++ b/documentation/proc-pages/physics-models/plasma_beta.md @@ -20,7 +20,7 @@ By default, $\beta$ is defined with respect to the total equilibrium B-field [^2 | 2 | Apply the $\beta$ limit to only the thermal plus neutral beam contributions to beta | | 3 | Apply the $\beta$ limit to the total beta (including the contribution from fast ions), calculated using only the toroidal field | -### Setting the beta $g$ coefficient +### Setting the Beta $g$ Coefficient Switch `iprofile` determines how the beta $g$ coefficient `dnbeta` should be calculated. @@ -28,11 +28,11 @@ be calculated. | `iprofile` | Description | | :-: | - | | 0 | `alphaj`, `rli` and `dnbeta` are inputs. | -| 1 | `alphaj`, `rli` and `dnbeta` are calulcated consistently. `dnbeta` calculated using $g=4l_i$ [^3]| +| 1 | `alphaj`, `rli` and `dnbeta` are calulcated consistently. `dnbeta` calculated using $g=4l_i$ [^3]. This is only recommended for high aspect ratio tokamaks.| | 2 | `alphaj` and `rli` are inputs. `dnbeta` calculated using $g=2.7(1+5\epsilon^{3.5})$ (which gives g = 3.0 for aspect ratio = 3) | | 3 | `alphaj` and `rli` are inputs. `dnbeta` calculated using $g=3.12+3.5\epsilon^{1.7}$ [^4]| -| 4 | `alphaj` and `dnbeta` are inputs. `rli` calculated from elongation [^4]| -| 5 | `alphaj` is an input inputs. `rli` calculated from elongation and `dnbeta` calculated using $g=3.12+3.5\epsilon^{1.7}$ [^4]| +| 4 | `alphaj` and `dnbeta` are inputs. `rli` calculated from elongation [^4]. This is only recommended for spherical tokamaks.| +| 5 | `alphaj` is an input. `rli` calculated from elongation and `dnbeta` calculated using $g=3.12+3.5\epsilon^{1.7}$ [^4]. This is only recommended for spherical tokamaks.| Further details on the calculation of `alphaj` and `rli` is given in [Plasma Current](./plasma_current.md). diff --git a/documentation/proc-pages/physics-models/plasma_current.md b/documentation/proc-pages/physics-models/plasma_current.md index 2be3309869..3a46d19b97 100644 --- a/documentation/proc-pages/physics-models/plasma_current.md +++ b/documentation/proc-pages/physics-models/plasma_current.md @@ -28,7 +28,7 @@ the switch `icurr`, as follows: A limited degree of self-consistency between the plasma current profile and other parameters [^6] can be enforced by setting switch `iprofile = 1`. This sets the current -profile peaking factor $\alpha_J$ (`alphaj`) and the normalised internal inductance $l_i$ (`rli`) using the +profile peaking factor $\alpha_J$ (`alphaj`), the normalised internal inductance $l_i$ (`rli`) and beta limit $g$-factor (`dnbeta`) using the safety factor on axis `q0` and the cylindrical safety factor $q*$ (`qstar`): $$\begin{aligned} @@ -46,7 +46,7 @@ g = 4 l_i It is recommended that current scaling law `icurr = 4` is used if `iprofile = 1`. This relation is only applicable to large aspect ratio tokamaks. -For spherical tokamaks, the internal inductance can be set from the elongation using `iprofile = 4` or `iprofile = 5`: +For spherical tokamaks, the normalised internal inductance can be set from the elongation using `iprofile = 4` or `iprofile = 5`: $$\begin{aligned} l_i = 3.4 - \kappa_x From 64b4f3eada35c3864310bec33a4539cb415d335f Mon Sep 17 00:00:00 2001 From: Stuart Muldrew Date: Tue, 23 Jul 2024 08:33:35 +0100 Subject: [PATCH 4/8] Added comment --- process/physics.py | 1 + 1 file changed, 1 insertion(+) diff --git a/process/physics.py b/process/physics.py index 9385cf8ce1..f691e90797 100644 --- a/process/physics.py +++ b/process/physics.py @@ -1402,6 +1402,7 @@ def physics(self): 3.12e0 + 3.5e0 * physics_variables.eps**1.7e0 ) + # culblm returns the betalim for beta physics_variables.betalim = culblm( physics_variables.bt, physics_variables.dnbeta, From b03715451302731f9e81acd86d465f504f1a2fc8 Mon Sep 17 00:00:00 2001 From: Stuart Muldrew Date: Tue, 23 Jul 2024 08:38:28 +0100 Subject: [PATCH 5/8] Ran black --- process/physics.py | 12 ++++-------- 1 file changed, 4 insertions(+), 8 deletions(-) diff --git a/process/physics.py b/process/physics.py index f691e90797..6b831e2c91 100644 --- a/process/physics.py +++ b/process/physics.py @@ -1398,9 +1398,7 @@ def physics(self): # Menard, et al. "Fusion Nuclear Science Facilities # and Pilot Plants Based on the Spherical Tokamak." # Nucl. Fusion, 2016, 44. - physics_variables.dnbeta = ( - 3.12e0 + 3.5e0 * physics_variables.eps**1.7e0 - ) + physics_variables.dnbeta = 3.12e0 + 3.5e0 * physics_variables.eps**1.7e0 # culblm returns the betalim for beta physics_variables.betalim = culblm( @@ -2155,20 +2153,18 @@ def culcur( icurr, plascur, q95, asp, eps, bt, kappa, triang, pperim, constants.rmu0 ) - - if iprofile == 1: # Ensure current profile consistency, if required # This is as described in Hartmann and Zohm only if icurr = 4 as well... - + # Tokamaks 4th Edition, Wesson, page 116 alphaj = qstar / q0 - 1.0 - rli = np.log(1.65 + 0.89 * alphaj) + rli = np.log(1.65 + 0.89 * alphaj) if iprofile == 4 or iprofile == 5: # Spherical Tokamak relation for internal inductance # Menard et al. (2016), Nuclear Fusion, 56, 106023 - rli = 3.4 - kappa + rli = 3.4 - kappa return alphaj, rli, bp, qstar, plascur From 458e34fbcacff4a3332b880ec50e2cf5f0892456 Mon Sep 17 00:00:00 2001 From: Stuart Muldrew Date: Tue, 23 Jul 2024 11:21:55 +0100 Subject: [PATCH 6/8] Updated beta equation in docs --- documentation/proc-pages/physics-models/plasma_beta.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/documentation/proc-pages/physics-models/plasma_beta.md b/documentation/proc-pages/physics-models/plasma_beta.md index 048d17f75f..05f295483f 100644 --- a/documentation/proc-pages/physics-models/plasma_beta.md +++ b/documentation/proc-pages/physics-models/plasma_beta.md @@ -3,7 +3,7 @@ The plasma beta limit[^1] is given by $$\begin{aligned} -\beta < g \, \frac{I(\mbox{MA})}{a(\mbox{m}) \, B_0(\mbox{T})} +\beta < 0.01\, g \, \frac{I(\mbox{MA})}{a(\mbox{m}) \, B_0(\mbox{T})} \end{aligned}$$ where $B_0$ is the axial vacuum toroidal field. The beta From 572e007644671f37293c58efb2f0cc94a50a1f37 Mon Sep 17 00:00:00 2001 From: Stuart Muldrew Date: Tue, 23 Jul 2024 15:43:10 +0100 Subject: [PATCH 7/8] Fixed typo in plasma_current.md --- documentation/proc-pages/physics-models/plasma_current.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/documentation/proc-pages/physics-models/plasma_current.md b/documentation/proc-pages/physics-models/plasma_current.md index 3a46d19b97..72d5a95a3d 100644 --- a/documentation/proc-pages/physics-models/plasma_current.md +++ b/documentation/proc-pages/physics-models/plasma_current.md @@ -111,7 +111,7 @@ Unpublished internal Oak Ridge document. Current Drive', ITER-TN-PH-8-4, 13--17 June 1988, Garching, FRG [^6]: Y. Sakamoto, 'Recent progress in vertical stability analysis in JA', Task meeting EU-JA #16, Fusion for Energy, Garching, 24--25 June 2014 -[^7] Menard et al. (2016), Nuclear Fusion, 56, 106023 +[^7]: Menard et al. (2016), Nuclear Fusion, 56, 106023 [^8]: H.R. Wilson, Nuclear Fusion **32** (1992) 257 [^9]: O. Sauter, C. Angioni and Y.R. Lin-Liu, Physics of Plasmas **6** (1999) 2834 [^10]: O. Sauter, C. Angioni and Y.R. Lin-Liu, Physics of Plasmas **9** (2002) 5140 \ No newline at end of file From 4f853544858a5d440ae271b19909af03997cc473 Mon Sep 17 00:00:00 2001 From: Stuart Muldrew Date: Tue, 23 Jul 2024 15:54:20 +0100 Subject: [PATCH 8/8] Fixed typo in plasma_current.md --- documentation/proc-pages/physics-models/plasma_current.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/documentation/proc-pages/physics-models/plasma_current.md b/documentation/proc-pages/physics-models/plasma_current.md index 72d5a95a3d..a1395a0edd 100644 --- a/documentation/proc-pages/physics-models/plasma_current.md +++ b/documentation/proc-pages/physics-models/plasma_current.md @@ -36,7 +36,7 @@ $$\begin{aligned} \end{aligned}$$ $$\begin{aligned} -l_i = ln(1.65+0.89\alpha_J) +l_i = \rm{ln}(1.65+0.89\alpha_J) \end{aligned}$$ $$\begin{aligned}