Evening Talk at EFTC-11 MAGNETIC GEOMETRY, PLASMA PROFILES AND STABILITY J W Connor UKAEA/EURATOM Fusion Association, Culham Science Centre, Abingdon, Oxon, OX14 3DB, UK A story.
Download ReportTranscript Evening Talk at EFTC-11 MAGNETIC GEOMETRY, PLASMA PROFILES AND STABILITY J W Connor UKAEA/EURATOM Fusion Association, Culham Science Centre, Abingdon, Oxon, OX14 3DB, UK A story.
Slide 1
Evening Talk at EFTC-11
MAGNETIC GEOMETRY, PLASMA PROFILES
AND STABILITY
J W Connor
UKAEA/EURATOM Fusion Association, Culham Science Centre,
Abingdon, Oxon, OX14 3DB, UK
A story about mode structures with the ‘Universal Mode’ providing a
‘Case History’
Slide 2
1. INTRODUCTION
The Universal Mode
● Anomalous transport associated with micro-instabilities such as the
electron drift wave
● Early theory for a homogeneous plasma slab or cylinder (with dn/dx =
constant) showed that in a shear free situation one could always find an
unstable mode
– driven by electron Landau resonance, with long parallel wavelengths
to minimise ion Landau damping
VT i
k ||
VT e
– the so-called Universal Mode
● This talk will explore how the stability and mode structure responds to
more realistic magnetic geometry and radial profiles
– leads to ballooning theory and more recent developments in this topic
Slide 3
Geometry
Cylinder
● n(r), q(r):
rB z
q =
RB
● Fourier analyse: = (r)e-i(m-nz/R)
●
k||
1
Rq
m nq(r)
● For electron Landau drive and to minimise ion Landau damping
VT i
● Shear
s
k ||
r dq
VT e
long parallel wavelength
0 mode localised around resonant surface
q dr
r0 : m = nq(r0)
k||
nq r - r0
r
Ls
,
1
Ls
s
Rq
Slide 4
Axisymmetric Torus, B(r,)
● = (r, )ein
● 2D (r,) – periodic in ; different poloidal m coupled
● High-n – simplify with eikonal
~
inS( r ,)
(r, ) ~ (r, )e
● Problem is to reconcile this with small k|| and periodicity
B S 0 S q(r )
but not periodic! ; shear q(r) q(r0)
Slide 5
Preview of Model Eigenvalue Equation
2
2
2
2
2 iX 1X 2X 2 cos issin i e (X, ) 0
X
X
FLR
Ion Sound
Radial variation
due to * or E
Toroidicity
Electron
Drive
Eigenvalue
()
- X = nq′ (r – r0)
•
Resonant surfaces
M
•
M+1
M+2
Different cases, depending on magnitudes of , 1, 2
M+3
X
Slide 6
2. SHEARED SLAB/CYLINDER
● Include magnetic shear, s 0, and density profile n(r)
● Eigenvalue equation
d2
2 2
2
X 2 X i e (X) () (X) 0
2
dX
FLR
shear
density
profile
electron Landau
drive
Potential Q(X)
● Treat e as perturbation
eigenvalue
Slide 7
MARSHALL ROSENBLUTH – 1927-2003
Slide 8
Low shear (or rapidly varying n(r))
● Localised in potential well (Krall-Rosenbluth)
– Im e : UNSTABLE
Increased shear
● Well becomes ‘hill’: no localised mode
– shear stabilisation criterion?
Ls
Ln
L
i
STABLE
Outgoing waves
● But, Pearlstein-Berk realised these are acceptable solutions (wave
propagates to ion Landau damping region where it becomes evanescent)!
e
ioX
2
, | X|
-i -
iL n
Ls
● Radiative or shear damping
Slide 9
Outgoing wave
(Ls/Ln)1/2
X
Q
● Balancing against e gives stability criterion
Ln
Ls
e STABLE
● ‘Confirmed’ by erroneous numerical calculations!
Ls/Ln
Potential ‘hill’
Slide 10
Aside
● But later numerical (Ross-Mahajan) and analytic (Tsang-Catto et al;
Antonsen, Liu Chen) treatments with full Plasma Dispersion function Z for
electrons showed
UNIVERSAL MODEL ALWAYS STABLE!
● Traced to fact that Z has a crucial effect on mode structure in narrow
region
X
me
mi
affecting e
perturbation treatment inadequate!
Slide 11
Quasi-modes
● In periodic cylinder
k|| m – nq (r)
● Fix n, different m have resonant surfaces rm: n = q(rm)
● For large n, m they are only separated by
X
r
~
1
nq
1
2
0( 1
n
2)
1
● Then each radially localised ‘m-mode’ ‘looks the same’ about its own
resonant surface
● Each mode has almost same frequency – ie almost degenerate and
satisfies
k ||
k
~
i
Ln
1
Slide 12
JOHN BRYAN TAYLOR
Slide 13
● Roberts and Taylor realised it was possible to superimpose them to form a
radially extended mode
– the twisted slicing or quasi-mode which maintains k|| << k
Slide 14
BORIS KADOMTSEV – 1928-1998
Slide 15
3. TOROIDAL GEOMETRY
● In a torus two new effects arise from inhomogeneous magnetic fields
– new drives from trapped electrons (Kadomtsev & Pogutse):
DESTABILISES universal mode: trapped particle modes
– changes in mode structure from magnetic drifts: affects shear
damping
●
B ~
B0
R
a vertical drift
Z
VD ~
mv
2
eBR
B
r
R
R0
Slide 16
● Doppler shift: - k . vD
T cos is sin
x
normal
;
T
r
1
R
geodesic
● (r) (r,): two-dimensional
k ||
1 i
i
m
nq
(
r
)
X
Rq
● Eigenvalue equation (absorb e into )
2
2
2
2
iX 2 cos is sin
2 X () 0
2
X
X
~ 0(1)
Slide 17
Suppression of Shear Damping (Taylor)
Reflect
X = 1
Xm
Xm+1
● Seek periodic solution m u m (X)e im
● Approximate translational invariance of rational surfaces 2 << 1
● Model equation: ignore geodesic drift
d2
2
2
(
X
m
)
u m (u m 1 u m 1 ) 0
2
dX
Slide 18
● Cylindrical limit: = 0
u m A me
i( Xm) / 2
2
– independent shear damped Fourier modes
● Strong toroidal coupling limit: > 1
um slowly varying ‘functions’ of m
u m 1 u m
u m Ae
du m
2
dm
1 d um
2 dm 2
( X m ) /
2
– each um is localised about x = m: no shear damping
iX /
~ Ae
2
– decays before ~ 2, so periodicity not an issue
Slide 19
● is a quasi-mode
– ‘balloons’ in
– ‘m’ varies with X
– radially extended
determine slowly varying radial envelope A(m) by reintroducing
2 << 1
– seen in gyrokinetic simulations (eg W Lee)
Slide 20
● Arbitrary
X
iXi kdX
Try
~ ()e
2
2
2
2
( k ) 2[cos s( k ) sin ] () 2 X 0
2
2 << - one-dimensional equation in
But solution must be periodic in over 2
- must reconcile with secular terms!
● Solve problem with Ballooning Transformation (Connor, Hastie, Taylor)
m
e
im
i ( X m k ) ~
de
(, X; k )
u m (X)
Slide 21
Aside
● The ballooning transformation was first developed for ideal MHD
ballooning modes, leading to the s- diagram
● Interesting to contrast published and actual diagrams!
● Lack of trust in
– numerical computation (confirmed by two-scale analysis for low s)
– ability to find access to 2nd stability (eg raise q0)
Slide 22
•
Alternatively (Lee, Van Dam)
Translational invariance limit, 2 0
system invariant under
XX+1 , mm+1
Bloch Theorem um(x) = eimk u(x – m)
Fourier Transform of u leads to ballooning equation
Slide 23
•
Return to drift wave:
2 2
2
2
( k ) 2 (cos s( k ) sin ) 2 X ( , X , k ) 0
2
●
Potential Q()
- < < no longer need periodic solution!
● Often consider ‘Lowest Order’ equation, 2 =0
– Schrődinger equation with potential Q() with k a parameter, () a
‘local’ eigenvalue
– k usually chosen to be 0 or (to give most unstable mode)
– increasing removes shear damping; vanishes for > 4: more
UNSTABLE
Slide 24
Potential Q
0
0
Marginal
s
-0.5
Shear damping
-1.0
-1.3
1.0
2.0
3.0
4.0
5.0
Slide 25
4. RADIAL MODE STRUCTURE
● ‘Higher order’ theory: determines radial envelope A(X) (Taylor, Connor,
Wilson)
● Reintroduce 1,2 << 1 k = k(X)
– yields radial envelope in WKB approximation
X
A( X) e
i k(X)dX
kdX
– eigenvalue condition
( 1 )
2
Conventional Version: * has a maximum
● Lowest order theory gives ‘local’ eigenvalue
2
* (0) 2 X
'local' * ( x )
i s (k )
shear damping
● Expand about k0 where shear damping is minimum
2
(0) 2 X i kk (k k 0 )
k (X, )
2
( kk ~ )
Slide 26
(a)
Quadratic profile
(b)
Linear profile
1/ 4
● Implications
a
1/ 2
– mode width X ~
~ n , r~ 1/2 a
M
n
2
– mode localised about X = 0, ie *max, , but covers many resonant
surfaces
– spread in k: k ~ n-1/2 << 1
k k0 (minimum shear damping)
Slide 27
Limitations
1. Low magnetic shear
2. High velocity shear
3. The edge
-
unfortunately characteristics of transport barriers!
ITB
H-mode
Slide 28
Low Magnetic Shear, s << 1
● Two-scale analysis of ballooning equations (, u): periodic
equilibrium scale, u = s
‘averaged’ eqn, independent of k independent of k!
– corresponds to uncoupled Fourier harmonics at each mth surface,
localised within X ~ |s|: ie non-overlapping radially
● Recover k-dependence by using these as trial functions in variational
approach (Romanelli and Zonca)
– exponentially weak contribution
2
2
* (0) 2 X i exp( c / | s |) ˆ kk (k k 0 )
1
4
s crit
XM ~
exp
|
s
|
2
– becomes very narrow as |s| 0, or ki 0
s crit (ki )
2
Slide 29
● Estimate anomalous transport: Lin XM2
suggests link between low
shear and ITBs
(Romanelli, Zonca)
scrit
1
s
● Presence of qmin acts as barrier to mode structures
Gyrokinetic simulation by Kishimoto
Slide 30
Modelling of Impurity Diffusion in JET with Dz ~ (iL)1/2 exp (-c/s), V = D/R
L Lauro-Taroni
Slide 31
● Ballooning mode theory fails for sufficiently low s (or long wavelength)
– reverts to weakly coupled Fourier harmonics, amplitude Am, when
ni
r
– spectrum of A m
Ln
2
4sq R
narrows as mode centre moves towards qmin
● In practice, ballooning theory holds to quite low n
eg ITG modes with kI ~ 0(1) largely unaffected by qmin
Slide 32
The Wavenumber Representation
● More general contours of k (X, )
(Romanelli, Zonca)
● ‘Closed’ contours already discussed; ‘passing’ contours sample all k
– WKB treatment in X-space still possible
– easier to use alternative, but entirely equivalent, Wavenumber
Representation (Dewar, Mahajan)
Slide 33
● (X, )
dk(, k) exp iX ( k ) S(k )
ˆ
● ˆ (, k ) satisfies ballooning eqn on - < < , ie not periodic in
ˆ ( 2, k 2)
ˆ (, k)
●
– is periodic in if S(k) is periodic in k: eigenvalue condition
Example
● Suppose linear profile: * (0) 1X i s (k )
dS i( (0) i (k )) 0
1
*
s
dk
● Periodicity of S yields eigenvalue condition
dk[ * i s (k)] 21
,
1 ~
1
~ 0
nqL
n
1
Slide 34
Slide 35
Implications
● Re related to local *(x):
● Im
*
2 1
1
dk s (k)
2
– k not restricted to near k0, all k contribute to give an average of the
shear damping!
– some shear damping restored: more STABLE
eg = 4: s(0) = - 0.02,
● k 2 X ~
1
1
(k ) ~
1
2
1
s (k)dk 0.35
1 if 1
● Mode width:
(i) real X ~ n, or r ~ a
(ii) complex X ~ n1/2 1/2, or r ~ (/n)1/2a
Slide 36
Sheared Radial Electric Fields
● Believed to reduce instability and turbulence – prominent near ITBs
● - n E (x) (Doppler Shift); suppose E = x
1 E
●
dE
dq
q ~ 0(1) !
X ~ / q
mode narrows as dE/dq increases, reducing estimates of X and
transport
● Are these modes related to conventional ballooning modes?
– introduce density profile variation
● Model: * (0) i 0 i xx X 2 i k cos k q X
ie has maximum at X = 0
Slide 37
● Wavenumber representation produces quadratic eqn for dS/dk
– exp (in S) (k)
- periodic S(k) (k ) is Floquet solution of Mathieu eqn: yields
eigenvalue
Analytic solution for transition region
possible (Connor)
crit
q
1 / 2
~ 0 T
n
continuous evolution from conventional mode to more STABLE
‘passing’ mode
●
X ~
(d E / dq )
1 for large dE/dq
reverts to Fourier modes!
FULL CIRCLE?
Slide 38
5. EXTENSIONS TO BALLOONING THEORY
● Have seen limitations imposed by low magnetic shear and high flow shear
● The presence of a plasma edge clearly breaks translational invariance
– have used 2D MHD code to study high-n edge ballooning modes;
mode structure resembles ballooning theory ‘prediction’
Slide 39
● Non-linear theory
– the ‘twisted slices’ of Roberts and Taylor form a basis for flux-tube
gyrokinetic simulations
y
x
Conventional
Tokamak
ST
Slide 40
– introducing non-linearities into the theory of high-n MHD ballooning
modes predicts explosively growing filamentary structures, seen on
MAST
Simulation
Experiment
Slide 41
6. SUMMARY AND CONCLUSIONS
● Have used the story of the ‘Universal Mode’ to illustrate developments in
the theory of toroidal mode structures and stability
– the universal mode is stable, but plenty of other toroidal modes are
available to provide anomalous transport!
● Problems of toroidal periodicity in the presence of magnetic shear
resolved by Ballooning theory
● Ballooning theory provides a robust and widely used tool, but its validity
can break down for:
– Low magnetic shear
– Rotation shear
– Plasma edge
when the higher order theory is considered
● Re-emergence of Fourier modes in the torus for low s and high dE/dq
● Ballooning theory also provides a basis for some non-linear theories and
simulations
Slide 42
UNSTABLE
Universal
Mode
(Galeev et al)
1963
Outgoing
Waves
(PearlsteinBerk 1969)
Loss of
shear
damping in
Torus
(Taylor) 1976
Toroidally Induced mode
(Chen-Cheng) 1980
(Hesketh, Hastie, Taylor)
1979
Time
1960
1970
Shear stabilisation
(Krall-Rosenbluth)
1965
STABLE
‘Correct’ calculation
(Tsang, Catto,
Ross, Mahajan:
Antonsen, Liu
Chen) 1978
2000
1990
1980
Steep
gradients
in torus
(Connor,
Taylor)
1987
Not at *MAX
(Connor,
Taylor,
Wilson)
1992, 1996
Low magnetic
shear
(Romanelli,
Zonca 1993;
Connor,
Hastie, 2003)
Velocity
shear
(Taylor,
Wilson;
Dewar)
1996-7
Evening Talk at EFTC-11
MAGNETIC GEOMETRY, PLASMA PROFILES
AND STABILITY
J W Connor
UKAEA/EURATOM Fusion Association, Culham Science Centre,
Abingdon, Oxon, OX14 3DB, UK
A story about mode structures with the ‘Universal Mode’ providing a
‘Case History’
Slide 2
1. INTRODUCTION
The Universal Mode
● Anomalous transport associated with micro-instabilities such as the
electron drift wave
● Early theory for a homogeneous plasma slab or cylinder (with dn/dx =
constant) showed that in a shear free situation one could always find an
unstable mode
– driven by electron Landau resonance, with long parallel wavelengths
to minimise ion Landau damping
VT i
k ||
VT e
– the so-called Universal Mode
● This talk will explore how the stability and mode structure responds to
more realistic magnetic geometry and radial profiles
– leads to ballooning theory and more recent developments in this topic
Slide 3
Geometry
Cylinder
● n(r), q(r):
rB z
q =
RB
● Fourier analyse: = (r)e-i(m-nz/R)
●
k||
1
Rq
m nq(r)
● For electron Landau drive and to minimise ion Landau damping
VT i
● Shear
s
k ||
r dq
VT e
long parallel wavelength
0 mode localised around resonant surface
q dr
r0 : m = nq(r0)
k||
nq r - r0
r
Ls
,
1
Ls
s
Rq
Slide 4
Axisymmetric Torus, B(r,)
● = (r, )ein
● 2D (r,) – periodic in ; different poloidal m coupled
● High-n – simplify with eikonal
~
inS( r ,)
(r, ) ~ (r, )e
● Problem is to reconcile this with small k|| and periodicity
B S 0 S q(r )
but not periodic! ; shear q(r) q(r0)
Slide 5
Preview of Model Eigenvalue Equation
2
2
2
2
2 iX 1X 2X 2 cos issin i e (X, ) 0
X
X
FLR
Ion Sound
Radial variation
due to * or E
Toroidicity
Electron
Drive
Eigenvalue
()
- X = nq′ (r – r0)
•
Resonant surfaces
M
•
M+1
M+2
Different cases, depending on magnitudes of , 1, 2
M+3
X
Slide 6
2. SHEARED SLAB/CYLINDER
● Include magnetic shear, s 0, and density profile n(r)
● Eigenvalue equation
d2
2 2
2
X 2 X i e (X) () (X) 0
2
dX
FLR
shear
density
profile
electron Landau
drive
Potential Q(X)
● Treat e as perturbation
eigenvalue
Slide 7
MARSHALL ROSENBLUTH – 1927-2003
Slide 8
Low shear (or rapidly varying n(r))
● Localised in potential well (Krall-Rosenbluth)
– Im e : UNSTABLE
Increased shear
● Well becomes ‘hill’: no localised mode
– shear stabilisation criterion?
Ls
Ln
L
i
STABLE
Outgoing waves
● But, Pearlstein-Berk realised these are acceptable solutions (wave
propagates to ion Landau damping region where it becomes evanescent)!
e
ioX
2
, | X|
-i -
iL n
Ls
● Radiative or shear damping
Slide 9
Outgoing wave
(Ls/Ln)1/2
X
Q
● Balancing against e gives stability criterion
Ln
Ls
e STABLE
● ‘Confirmed’ by erroneous numerical calculations!
Ls/Ln
Potential ‘hill’
Slide 10
Aside
● But later numerical (Ross-Mahajan) and analytic (Tsang-Catto et al;
Antonsen, Liu Chen) treatments with full Plasma Dispersion function Z for
electrons showed
UNIVERSAL MODEL ALWAYS STABLE!
● Traced to fact that Z has a crucial effect on mode structure in narrow
region
X
me
mi
affecting e
perturbation treatment inadequate!
Slide 11
Quasi-modes
● In periodic cylinder
k|| m – nq (r)
● Fix n, different m have resonant surfaces rm: n = q(rm)
● For large n, m they are only separated by
X
r
~
1
nq
1
2
0( 1
n
2)
1
● Then each radially localised ‘m-mode’ ‘looks the same’ about its own
resonant surface
● Each mode has almost same frequency – ie almost degenerate and
satisfies
k ||
k
~
i
Ln
1
Slide 12
JOHN BRYAN TAYLOR
Slide 13
● Roberts and Taylor realised it was possible to superimpose them to form a
radially extended mode
– the twisted slicing or quasi-mode which maintains k|| << k
Slide 14
BORIS KADOMTSEV – 1928-1998
Slide 15
3. TOROIDAL GEOMETRY
● In a torus two new effects arise from inhomogeneous magnetic fields
– new drives from trapped electrons (Kadomtsev & Pogutse):
DESTABILISES universal mode: trapped particle modes
– changes in mode structure from magnetic drifts: affects shear
damping
●
B ~
B0
R
a vertical drift
Z
VD ~
mv
2
eBR
B
r
R
R0
Slide 16
● Doppler shift: - k . vD
T cos is sin
x
normal
;
T
r
1
R
geodesic
● (r) (r,): two-dimensional
k ||
1 i
i
m
nq
(
r
)
X
Rq
● Eigenvalue equation (absorb e into )
2
2
2
2
iX 2 cos is sin
2 X () 0
2
X
X
~ 0(1)
Slide 17
Suppression of Shear Damping (Taylor)
Reflect
X = 1
Xm
Xm+1
● Seek periodic solution m u m (X)e im
● Approximate translational invariance of rational surfaces 2 << 1
● Model equation: ignore geodesic drift
d2
2
2
(
X
m
)
u m (u m 1 u m 1 ) 0
2
dX
Slide 18
● Cylindrical limit: = 0
u m A me
i( Xm) / 2
2
– independent shear damped Fourier modes
● Strong toroidal coupling limit: > 1
um slowly varying ‘functions’ of m
u m 1 u m
u m Ae
du m
2
dm
1 d um
2 dm 2
( X m ) /
2
– each um is localised about x = m: no shear damping
iX /
~ Ae
2
– decays before ~ 2, so periodicity not an issue
Slide 19
● is a quasi-mode
– ‘balloons’ in
– ‘m’ varies with X
– radially extended
determine slowly varying radial envelope A(m) by reintroducing
2 << 1
– seen in gyrokinetic simulations (eg W Lee)
Slide 20
● Arbitrary
X
iXi kdX
Try
~ ()e
2
2
2
2
( k ) 2[cos s( k ) sin ] () 2 X 0
2
2 << - one-dimensional equation in
But solution must be periodic in over 2
- must reconcile with secular terms!
● Solve problem with Ballooning Transformation (Connor, Hastie, Taylor)
m
e
im
i ( X m k ) ~
de
(, X; k )
u m (X)
Slide 21
Aside
● The ballooning transformation was first developed for ideal MHD
ballooning modes, leading to the s- diagram
● Interesting to contrast published and actual diagrams!
● Lack of trust in
– numerical computation (confirmed by two-scale analysis for low s)
– ability to find access to 2nd stability (eg raise q0)
Slide 22
•
Alternatively (Lee, Van Dam)
Translational invariance limit, 2 0
system invariant under
XX+1 , mm+1
Bloch Theorem um(x) = eimk u(x – m)
Fourier Transform of u leads to ballooning equation
Slide 23
•
Return to drift wave:
2 2
2
2
( k ) 2 (cos s( k ) sin ) 2 X ( , X , k ) 0
2
●
Potential Q()
- < < no longer need periodic solution!
● Often consider ‘Lowest Order’ equation, 2 =0
– Schrődinger equation with potential Q() with k a parameter, () a
‘local’ eigenvalue
– k usually chosen to be 0 or (to give most unstable mode)
– increasing removes shear damping; vanishes for > 4: more
UNSTABLE
Slide 24
Potential Q
0
0
Marginal
s
-0.5
Shear damping
-1.0
-1.3
1.0
2.0
3.0
4.0
5.0
Slide 25
4. RADIAL MODE STRUCTURE
● ‘Higher order’ theory: determines radial envelope A(X) (Taylor, Connor,
Wilson)
● Reintroduce 1,2 << 1 k = k(X)
– yields radial envelope in WKB approximation
X
A( X) e
i k(X)dX
kdX
– eigenvalue condition
( 1 )
2
Conventional Version: * has a maximum
● Lowest order theory gives ‘local’ eigenvalue
2
* (0) 2 X
'local' * ( x )
i s (k )
shear damping
● Expand about k0 where shear damping is minimum
2
(0) 2 X i kk (k k 0 )
k (X, )
2
( kk ~ )
Slide 26
(a)
Quadratic profile
(b)
Linear profile
1/ 4
● Implications
a
1/ 2
– mode width X ~
~ n , r~ 1/2 a
M
n
2
– mode localised about X = 0, ie *max, , but covers many resonant
surfaces
– spread in k: k ~ n-1/2 << 1
k k0 (minimum shear damping)
Slide 27
Limitations
1. Low magnetic shear
2. High velocity shear
3. The edge
-
unfortunately characteristics of transport barriers!
ITB
H-mode
Slide 28
Low Magnetic Shear, s << 1
● Two-scale analysis of ballooning equations (, u): periodic
equilibrium scale, u = s
‘averaged’ eqn, independent of k independent of k!
– corresponds to uncoupled Fourier harmonics at each mth surface,
localised within X ~ |s|: ie non-overlapping radially
● Recover k-dependence by using these as trial functions in variational
approach (Romanelli and Zonca)
– exponentially weak contribution
2
2
* (0) 2 X i exp( c / | s |) ˆ kk (k k 0 )
1
4
s crit
XM ~
exp
|
s
|
2
– becomes very narrow as |s| 0, or ki 0
s crit (ki )
2
Slide 29
● Estimate anomalous transport: Lin XM2
suggests link between low
shear and ITBs
(Romanelli, Zonca)
scrit
1
s
● Presence of qmin acts as barrier to mode structures
Gyrokinetic simulation by Kishimoto
Slide 30
Modelling of Impurity Diffusion in JET with Dz ~ (iL)1/2 exp (-c/s), V = D/R
L Lauro-Taroni
Slide 31
● Ballooning mode theory fails for sufficiently low s (or long wavelength)
– reverts to weakly coupled Fourier harmonics, amplitude Am, when
ni
r
– spectrum of A m
Ln
2
4sq R
narrows as mode centre moves towards qmin
● In practice, ballooning theory holds to quite low n
eg ITG modes with kI ~ 0(1) largely unaffected by qmin
Slide 32
The Wavenumber Representation
● More general contours of k (X, )
(Romanelli, Zonca)
● ‘Closed’ contours already discussed; ‘passing’ contours sample all k
– WKB treatment in X-space still possible
– easier to use alternative, but entirely equivalent, Wavenumber
Representation (Dewar, Mahajan)
Slide 33
● (X, )
dk(, k) exp iX ( k ) S(k )
ˆ
● ˆ (, k ) satisfies ballooning eqn on - < < , ie not periodic in
ˆ ( 2, k 2)
ˆ (, k)
●
– is periodic in if S(k) is periodic in k: eigenvalue condition
Example
● Suppose linear profile: * (0) 1X i s (k )
dS i( (0) i (k )) 0
1
*
s
dk
● Periodicity of S yields eigenvalue condition
dk[ * i s (k)] 21
,
1 ~
1
~ 0
nqL
n
1
Slide 34
Slide 35
Implications
● Re related to local *(x):
● Im
*
2 1
1
dk s (k)
2
– k not restricted to near k0, all k contribute to give an average of the
shear damping!
– some shear damping restored: more STABLE
eg = 4: s(0) = - 0.02,
● k 2 X ~
1
1
(k ) ~
1
2
1
s (k)dk 0.35
1 if 1
● Mode width:
(i) real X ~ n, or r ~ a
(ii) complex X ~ n1/2 1/2, or r ~ (/n)1/2a
Slide 36
Sheared Radial Electric Fields
● Believed to reduce instability and turbulence – prominent near ITBs
● - n E (x) (Doppler Shift); suppose E = x
1 E
●
dE
dq
q ~ 0(1) !
X ~ / q
mode narrows as dE/dq increases, reducing estimates of X and
transport
● Are these modes related to conventional ballooning modes?
– introduce density profile variation
● Model: * (0) i 0 i xx X 2 i k cos k q X
ie has maximum at X = 0
Slide 37
● Wavenumber representation produces quadratic eqn for dS/dk
– exp (in S) (k)
- periodic S(k) (k ) is Floquet solution of Mathieu eqn: yields
eigenvalue
Analytic solution for transition region
possible (Connor)
crit
q
1 / 2
~ 0 T
n
continuous evolution from conventional mode to more STABLE
‘passing’ mode
●
X ~
(d E / dq )
1 for large dE/dq
reverts to Fourier modes!
FULL CIRCLE?
Slide 38
5. EXTENSIONS TO BALLOONING THEORY
● Have seen limitations imposed by low magnetic shear and high flow shear
● The presence of a plasma edge clearly breaks translational invariance
– have used 2D MHD code to study high-n edge ballooning modes;
mode structure resembles ballooning theory ‘prediction’
Slide 39
● Non-linear theory
– the ‘twisted slices’ of Roberts and Taylor form a basis for flux-tube
gyrokinetic simulations
y
x
Conventional
Tokamak
ST
Slide 40
– introducing non-linearities into the theory of high-n MHD ballooning
modes predicts explosively growing filamentary structures, seen on
MAST
Simulation
Experiment
Slide 41
6. SUMMARY AND CONCLUSIONS
● Have used the story of the ‘Universal Mode’ to illustrate developments in
the theory of toroidal mode structures and stability
– the universal mode is stable, but plenty of other toroidal modes are
available to provide anomalous transport!
● Problems of toroidal periodicity in the presence of magnetic shear
resolved by Ballooning theory
● Ballooning theory provides a robust and widely used tool, but its validity
can break down for:
– Low magnetic shear
– Rotation shear
– Plasma edge
when the higher order theory is considered
● Re-emergence of Fourier modes in the torus for low s and high dE/dq
● Ballooning theory also provides a basis for some non-linear theories and
simulations
Slide 42
UNSTABLE
Universal
Mode
(Galeev et al)
1963
Outgoing
Waves
(PearlsteinBerk 1969)
Loss of
shear
damping in
Torus
(Taylor) 1976
Toroidally Induced mode
(Chen-Cheng) 1980
(Hesketh, Hastie, Taylor)
1979
Time
1960
1970
Shear stabilisation
(Krall-Rosenbluth)
1965
STABLE
‘Correct’ calculation
(Tsang, Catto,
Ross, Mahajan:
Antonsen, Liu
Chen) 1978
2000
1990
1980
Steep
gradients
in torus
(Connor,
Taylor)
1987
Not at *MAX
(Connor,
Taylor,
Wilson)
1992, 1996
Low magnetic
shear
(Romanelli,
Zonca 1993;
Connor,
Hastie, 2003)
Velocity
shear
(Taylor,
Wilson;
Dewar)
1996-7