1. Introduction
The transport barrier (TB) concept was introduced to fluid mechanics (FM) by Prandtl as the laminar/turbulent boundary layer (LBL/TBL) [
1,
2,
3], where the LBL constitutes an edge transport barrier (ETB) using nomenclature from magnetic confinement fusion plasma physics (PP). The boundary layer (BL) is characterised by mean velocity shear and molecular (LBL)/turbulent (TBL) viscosity. The ETB was first identified in magnetically confined fusion plasmas in 1982 [
4] and named the high (H) confinement mode as opposed to the previously known low (L) confinement mode.
In 1995, uniform momentum zones (UMZs), regions where the streamwise momentum is close to being constant, were discovered [
5]. The UMZs are separated by internal shear layers. That year, internal transport barriers (ITBs) were discovered in two magnetically confined fusion experiments [
6,
7].
In this review, we will attempt to link the LBL and the ETB concepts and the UMZ and ITB concepts  for the first time to the best of our knowledge. Probably wall and magneticallybounded turbulent flows have not been systematically associated for several reasons, e.g. (i) because the PP community first and foremost focuses on links with astrophysical plasmas (magnetohydrodynamic (MHD) phenomena) and (ii) the FM community has not been considering parallel efforts in the PP community in a systematic fashion. However, FM research, for example the Kolmogorov 1941 (K41) energy cascade [
8,
9], has been used for PP turbulence studies. Other examples are covered in e.g. [
10,
11]. In addition to the LBL/ETB and UMZ/ITB similarities, other observations which triggered this review include:
Increasing core fluctuations for the pipe flow high Reynolds number (
$Re$) transition [
12] is similar to controlled confinement transitions in fusion plasmas [
13,
14]
Travelling wave solutions in pipe flow [
15] are reminiscent of the magnetic field structure (islands) in fusion plasmas
This review is personal in the sense that it is a result of my own "voyage through turbulence" [
3], which began in academic PP research (19972005), where I also had contact with K41 and energy/enstrophy cascades for two and threedimensional flows. Another major influence was the KolmogorovArnoldMoser (KAM) theorem [
16] and the survival or destruction of invariant tori in response to perturbations, which is directly applicable to the magnetic field structure. Moving to industrial FM research (20062023), my focus was on wallbounded turbulent flows, e.g. turbulent mixing of gases, twophase flow, flow noise (acoustics) and thermofluids. This is an attempt to synthesise my experience, but there is a risk of not referring to the latest research; particularly for PP, since I have not been active in the field since 2005. Wall and magneticallybounded flows are treated, but crossdisciplinary efforts have also included unbounded flows, e.g. similarities between turbulence on mm and Mpc scales [
17,
18]. However, those results are outside the scope of the present review.
In order to avoid copyright issues, figures from cited papers will be discussed but not shown. This unfortunately makes the review more difficult to read, but open source versions of most references can be found online.
The review is organised as follows:
Section 2 and
Section 3 consist of primers (in the spirit of The Los Alamos Primer [
19]) on wall and magneticallybounded turbulent flows, respectively. In
Section 4 transport barriers are treated in general and
Section 5 focuses on comparing turbulent flows in the core region. An overview of important concept similarities and differences follows in
Section 6. A discussion can be found in
Section 7 and we conclude in
Section 8.
2. Wallbounded turbulent flows
There are two main ways to treat wallbounded turbulent flows; one is the statistical approach and the other is a dynamical systems viewpoint [
20]. An important difference is that the (traditional) statistical approach considers turbulent flows with high
$Re$, whereas the dynamical systems analysis is limited to lower
$Re$. We will focus on the statistical point of view below but will discuss the dynamical systems approach in
Section 2.16. Research on the laminarturbulent pipe flow transition [
21] identifies a third perspective, which is linear or nonlinear hydrodynamic stability. This has been deemed out of scope for this review and will not be covered.
Canonical, i.e. standard, wallbounded flows include zero pressure gradient (ZPG) TBLs, channels and pipes [
22]. In the following, we focus on pipe flow but will also address features of other canonical flows.
The coordinates are usually named as (i) streamwise (x along the flow), (ii) wallnormal (y perpendicular to the wall) and (iii) spanwise (z parallel to the wall and perpendicular to the streamwise direction).
We assume the noslip and nopenetration boundary conditions (BCs) [
23], i.e. that the velocity at the wall is zero and that the walls are impermeable.
2.1. Transition from laminar to turbulent flow
To define the bulk Reynolds number
$R{e}_{D}$, where
$D=2R$ is the pipe diameter and
R is the pipe radius, the areaaveraged streamwise mean flow velocity
${U}_{m}$ is used:
where
${\nu}_{\mathrm{kin}}$ is the kinematic viscosity.
At a certain
$R{e}_{D}$ (
$\sim 2000$), the laminar to turbulent transition takes place [
3,
24], associated with a steepening of the edge velocity gradient. However, the transition is gradual with
$R{e}_{D}$; as it increases, what is observed first are turbulent puffs, which are regions of turbulence separated by laminar regions. Turbulent puffs either decay or split, both with very long timescales. As
$R{e}_{D}$ increases further, the turbulent patches increase in size and become what is called slugs, before turbulent flow fills the entire pipe [
21].
2.2. The boundary layer concept
Both velocity (momentum) and temperature (heat) BLs exist in wallbounded laminar and turbulent flows [
2]. The concepts are analogous, with a region of (velocity/temperature) gradients close to the wall and another region of (almost) constant values towards the pipe axis. The thermal BL can either be coupled to the velocity field or not depending on the conditions, e.g. assumptions on density, dynamic viscosity, specific heat capacity and thermal conductivity.
2.3. The turbulent/nonturbulent interface
In addition to a BL close to the wall, TBLs also have a turbulent/nonturbulent interface (TNTI) at the freestream boundary where the TBL ends [
25,
26,
27].
The TNTI was identified in [
25] from experiments and characterised as a thin fluid layer where viscous forces dominate, the "laminar superlayer", thought to be a wrinkled sheet of viscous vortical fluid. The mean and fluctuating vorticity propagate through this (wrinkled) layer to the nonturbulent (irrotational) region. The thickness of the layer is found to be of the order of the Kolmogorov length:
where
$\epsilon $ is the dissipation rate of
k, the turbulent kinetic energy (TKE) per unit mass.
Direct numerical simulations (DNS) of TBLs were presented in [
26], where a small peak in the spanwise vorticity and an associated small jump in streamwise velocity was observed at the TNTI. The interfacial layer was found to have an inertiaviscous double structure:
The length scale of the turbulent sublayer ${l}_{I}$ is longer than the length scale of the outer boundary ${l}_{S}$.
Analysis shows that the TNTI acts as a barrier in both directions: Exterior irrotational fluctuations are being damped/filtered at the interface and internal rotational fluctuations are also blocked at the TNTI which remains sharp.
Velocity jumps at the TNTI and inside the TBL were studied experimentally in [
27] and found to have similar characteristics. The velocity jump height was found to be constant for
$y/{\delta}_{99}>0.5$, i.e. far from the wall, with larger jumps closer to the wall. Here,
${\delta}_{99}$ is the (99%) TBL width, where
$\delta $ corresponds to
R in a pipe. The internal layers are regions of high shear which are thought to bound large scale motions (LSM), see
Section 2.7. The jump thickness
${\delta}_{w}$ is observed to scale with the (local) Taylor microscale:
${\delta}_{w}\approx 0.4{\lambda}_{T}$. The internal layers are observed to move away from the wall, with a faster layer velocity further from the wall. It is conjectured that shear layers are generated not only at the wall, but away from the wall as well.
2.4. Mean turbulent flow
The mean flow is in the streamwise direction, with three main wallnormal regions: the viscous sublayer closest to the wall, the logarithmic (log) layer and the wake region towards the pipe axis [
28,
29]. Sometimes the terms inner (outer) layer are used for the regions close to (far away) from the wall, respectively.
2.5. Fluctuating turbulent flow
Streamwise velocity fluctuations have a peak close to the wall (the inner peak) and a second peak in the log region which becomes more prominent with increasing
$Re$ (the outer peak). The inner peak has fixed wallnormal position (normalised to the viscous length scale), but it is under discussion if it has a maximum or continues to increase with
$Re$. The attached eddy model (AEM) [
30,
31] leads to structures increasing in size from the wall towards the pipe axis, also as a (streamwise and spanwise, but not wallnormal) loglaw, but decreasing towards the pipe axis as opposed to the mean streamwise flow [
32].
Streamwise velocity fluctuations are usually higher than both the wallnormal and spanwise fluctuations; energy transfer takes places from the streamwise to the wallnormal and spanwise fluctuations [
2].
2.6. Turbulence models
Turbulence models attempt to close the equations of motion, e.g. by introducing a turbulent (eddy) viscosity; for the simplest algebraic model, the turbulent viscosity is proportional to the mixing length, which is a concept introduced by Prandtl [
33,
34]. The turbulent shear (streamwise/wallnormal) Reynolds stress (RS)
${\tau}_{xy}$ is then equal to the product of the dynamic turbulent viscosity
${\mu}_{t}$ and the mean velocity gradient
$\mathcal{S}=\partial U/\partial y$:
where
${n}_{f}$ is the fluid density and
${\nu}_{t}$ is the kinematic turbulent viscosity. The turbulent RS represents the turbulent transport of momentum to the wall.
2.7. Turbulent structures
Turbulence consists of smaller structures in the inner layer, whereas both small and large structures coexist in the outer layer. The structures can be sorted into four different groups [
35]:
Sublayer (nearwall) streaks generated by streamwise vortices [
21]
Hairpin or $\Lambda $ vortices
Vortex packets or LSM
Even larger structures, called (i) very large scale motions (VLSM) in pipe flow and (ii) superstructures in boundary layers
The hairpin or $\Lambda $ vortices are vorticity structures with a "head" and two "feet"; the head is typically further downstream than the feet, i.e. the vortices are leaning in the streamwise direction.
There is an ongoing discussion on the interaction between structures  whether large structures in the outer layer are superimposed onto inner layer structures or if the mechanism is amplitude modulation [
36,
37]. There is also a discussion whether the large structures are "active" or "passive", i.e. whether they contribute to the turbulent shear RS or not [
38].
An area of research that has traditionally been included in the statistical approach but which also contains elements of the dynamical systems viewpoint is proper orthogonal decomposition [
39], which has e.g. been used to analyse radial and azimuthal modes of VLSM [
35].
2.8. Minimal flow unit
A minimal flow unit (MFU) has been identified [
40] which is a minimum structure size needed to sustain smallscale turbulence close to the wall. This has been done using DNS to isolate small structures in the inner layer.
The spanwise MFU ${\lambda}_{z}^{+}={\lambda}_{z}{u}_{\tau}/{\nu}_{\mathrm{kin}}\approx 100$, where "+" indicates normalisation by the viscous length scale ${\nu}_{\mathrm{kin}}/{u}_{\tau}$. Here, ${u}_{\tau}$ is the friction velocity. The spanwise MFU matches the value widely observed for the spacing of sublayer streaks and streamwise vortices. The streamwise MFU was observed to be ${\lambda}_{x}^{+}\approx 250350$, which is of the same order as experimental observations of vortices near a wall. Turbulence statistics are in good agreement with simulations covering the entire crosssection below a wallnormal distance ${y}^{+}=40$; nearwall turbulence can be sustained indefinitely for a layer width of this size.
Subsequent work on MFUs [
41] found two different streamwise MFUs:
2.9. Turbulent length scales
We have already introduced the Kolmogorov and Taylor length scales in
Section 2.3. Two other useful scales can be added, the first being the mixing length (mentioned in
Section 2.6):
and the second being the length scale of larger eddies:
see [
12] and the associated Supplementary Information for more details.
The Kolmogorov scale is the smallest scale and L is the largest scale. The Taylor and mixing length scales are intermediate (meso), with the Taylor length being shorter than the mixing length.
For the loglaw region we can write:
and define a length scale associate with the mean velocity gradient:
which can be used to rewrite the mean velocity gradient as:
Two other length scales have also previously been mentioned, the largest (outer) scale
$\delta $ (or
R) in
Section 2.3 and the small (inner) viscous length scale in
Section 2.8. The ratio between these scales defines the friction Reynolds number:
From these length scales, it has been argued that mixed scaling can be relevant, i.e. combinations of the inner and outer length scales, for example [
23]:
2.10. Uniform momentum zones
The first type of internal TBL observed was the UMZ, with nearly constant streamwise momentum separated by thin viscousinertial shear layers [
5]. In the shear layers, spanwise vorticity is lumped into strongly vortical regions, i.e. a collection of vortices. This interpretation differs from the picture in [
25], where the TNTI was interpreted as a continuous vortex sheet.
Later observations in TBLs have continued to study the UMZ structure and the intense vorticity in the shear layers [
42]. The number of UMZs increases proportionally to
$log\left(R{e}_{\tau}\right)$ and the UMZ thickness increases with increasing distance from the wall. The structures generating the UMZ behave consistently with the AEM: Hairpin packets are shown to create a zonallike organisation.
An UMZ vortical fissure (VF) model was presented in [
43] and validated against DNS simulations of channel flow. The UMZs are segregated by narrow fissures of concentrated vorticity, with a discrete number of fissures (internal shear layers) across the TBL. The model has two primary domains, (i) an inertial domain and (ii) a subinertial domain; the theoretical basis for the inertial layer (far from wall) is more solid than for the subinertial layer (nearwall). A fixed fissure width gives the best match to DNS and the jump in streamwise velocity is proportional to
${u}_{\tau}$. The wake is not taken into account for the modelled mean velocity. The internal VFs are allowed to be repositioned (from an initial master profile) and a momentumexchange mechanism is necessary:
Outward flux of vorticity is connected with inward flux of momentum

The VF characteristic velocity is recalculated:
 
If the VF moves farther from (toward) the wall, there is momentum loss (gain) compared to the master profile
The outermost VF is not allowed to move and exchange momentum
The momentumexchange mechanism, i.e. that VFs gain (lose) momentum when they are displaced toward (away) from the wall, is consistent with a variation of the streamwise/wallnormal turbulent RS:
where overbar is time averaging and
${\omega}_{z}$ is the spanwise vorticity. The last righthand side term is zero because only wallnormal VF movement is considered.
2.11. Quiescent core
For turbulent channel flow, what is known as the quiescent core has been identified and characterised [
44]. The quiescent core is a large UMZ, which can cover up to 4045% of the channel; it can be approximated by regions where the mean velocity is above 95 % of the centerline (CL) mean velocity:
$U>0.95{U}_{CL}$. The interface has a jump in streamwise velocity, and sometimes  but not always  a vorticity peak. Inside the core UMZ, the streamwise velocity varies only weakly. The core UMZ is meandering (moves around), can reach the wall and be streamwise separated (breakup). The core UMZ has low TKE, i.e. it is weakly turbulent (quiescent).
A twostate model of the TBL (extendable to internal flows) is presented in [
45] to capture the loglaw and law of the wake regions. The new model has a loglaw state and a free stream state, with a velocity jump at their interface. The concept for mean flow can be applied to streamwise turbulence as well. One drawback of the model is that it does not take the viscous region close to the wall into account. The position of the interface is fitted to a Gaussian distribution which is independent of
$R{e}_{\tau}$. The resulting velocity jumps and deviations of the fit from the loglaw are also independent of
$R{e}_{\tau}$ except for pipe flow below
$R{e}_{\tau}=3400$, which is interesting and may be related to the high Reynolds number transition region for pipe flow [
12].
Open channel flow was studied in [
46], and it was concluded that: "The virtual absence of a wake region and of corrective terms to the loglaw in the present flow leads us to conclude that deviations from the loglaw observed in internal flows are likely due to the effects of the opposing walls, rather than the presence of a driving pressure gradient." Thus, the law of the wake may only exist due to TBL interactions.
2.12. Uniform temperature zones
After the identification of UMZ, uniform thermal zones (UTZ) have been found, which consist of regions of relatively uniform temperature separated by thermal interface layers [
47]. The analysis was done on DNS simulations of transcritical channel flow. An analogy was made between UMZ and momentum internal interface layers (MIILs) and UTZ and thermal internal interface layers (TIILs). Thus, the two types of zones relate to velocity (momentum) and temperature (heat) fields. A local heat transfer peak is expected in the TIILs. The MIILs and TIILs were found to be at similar but not identical locations, i.e. not collocated.
A model of UTZ and TIILs has been published in [
48], constructed along the same lines as the UMZ model in [
43]. The nomenclature is slightly modified compared to [
47]; here, the uniform zones are called uniform temperature zones and the TIILs are named thermal fissures (TF). The heat model (UTZ/TF) is combined with the momentum model (UMZ/VF) and calibrated against DNS simulations of channel flow. As for the momentum model, the TFs can move (from an original master profile) and exchange heat as they move in the wallnormal direction: If a TF moves towards (away from) the wall, its temperature increases (decreases), respectively. The finding in [
47] that the VF/TF (MIIL/TIIL) are correlated but not coincident is confirmed in [
48].
It is important to note that temperature is a passive scalar [
49] (when buoyancy is neglected), i.e. it does not affect the dynamics of the fluid.
2.13. Uniform concentration zones
The third type of uniform zone (UZ) reported is uniform concentration zones (UCZ) [
50]: As is the case for temperature, concentration is also a passive scalar.
In both shear and shearfree flows, rampcliff (RC) structures have been identified for passive scalars, i.e. a slow increase (ramp) followed by a fast decrease (cliff) [
51]. These structures have also been said to have a "sawtooth appearance" with plateaus separated by cliffs [
52]. The RC structures can be understood as large counterrotating structures which form a saddle point associated with convergingdiverging separatrices as discussed in [
53]. The cliff (or front) occurs at the diverging separatrix, which has an inclination close to the direction of the principal axis of strain. If the passive scalar is temperature, the front is the separation between warm and cold fluids entrained in the counterflowing structures. In atmospheric flow, clifframp (CR) structures have been considered signatures of the KelvinHelmholtz instability [
54].
2.14. Uniform momentum and temperature zones
Simultaneous existence of both UMZ and UTZ has been reported for both stably and unstably stratified turbulent flow by analysis of large eddy simulations (LES) [
55,
56]. In [
55] the stably stratified planetary boundary layer (PBL) was treated; it was found that UMZ and UTZ are "closely, but not perfectly related". Unstably stratified channel flow was covered in [
56], where it was found that: "Conditional averaging indicates that both UMZ and UTZ interfaces are associated with ejections of momentum and warm updrafts below the interface and sweeps of momentum and cool downdrafts above the interface."
2.15. Turbulence control
Methods for classical flow control up to around the year 2000 have been covered in [
57]. Methods can be active or passive, e.g.:
Passive: Riblets, surface treatment, tripping, shaping
Active: Suction, blowing, wall cooling/heating
Here, the purpose can be e.g. to modify transition to turbulence, to decrease friction (pressure drop), to enhance heat transfer and to reduce acoustic noise [
58].
More recent work includes turbulence suppression due to pulsatile driving of pipe flow [
59]. The work was inspired by the human cardiovascular system, where blood flow in the aorta is an example of pulsating flow. It is demonstrated that both turbulence and turbulent drag can be reduced significantly in pulsating flow.
Machine learning (ML) has in recent years become a more powerful tool for both turbulence simulation and control [
60]. The method can be seen as a fourth pillar complementing theory, experiments and simulations.
2.16. Dynamical systems viewpoint
For the dynamical systems approach, we focus on invariant solutions to the NavierStokes equations (NSE) as defined in [
61]:
"Here by ‘invariant solutions’ or ‘exact coherent structures’ we mean compact, timeinvariant solutions that are setwise invariant under the time evolution and the continuous symmetries of the dynamics. Invariant solutions include, for instance, equilibria, travelling waves, periodic orbits and invariant tori. Note in particular that the closure of a relative periodic orbit is an invariant torus."
The first exact coherent state (ECS) or travelling wave (TW) solution to the NSE was identified theoretically in [
62] followed by multiple efforts, both with theoretical [
15,
63,
64,
65,
66] and experimental [
20,
67,
68] focus.
For pipe flow, it has been found that the ECS originate in saddlenode bifurcations at
$R{e}_{D}$ down to around 400 [
69]. The TWs consist of a certain number of azimuthally and radially separated streaks, for example threefold azimuthal symmetry: 6 outer (high speed) streaks and 3 inner (low speed) streaks. The TWs lead to transport of slow fluid towards the center and transport of fast fluid towards the wall.
Additional TW solutions were constructed in [
15] by "mixing three key flow structures  2dimensional streamwise rolls, streaks and 3dimensional streamwisedependent waves  in the right way". This is in line with what has been termed the selfsustaining process (SSP), see [
63] and references therein. Here, it is proposed that turbulence is maintained (against viscosity) by a cycle of rolls, streaks and waves.
A main obstacle to a direct link between the dynamical and the statistical approach is to identify invariant solutions for high
$Re$. Experimental support that these solutions exist have come from [
20], where ECS are shown to have an impact up to
$R{e}_{D}=35000$.
Other ECS solutions have been investigated in parallel, we refer to related work focusing on the relative periodic orbit (RPO) framework [
61,
70,
71,
72]. The two types of ECS solutions can be summarised as:
TW: A fixed velocity profile moving in the streamwise direction with a constant phase speed
RPO: Timedependent velocity profiles which repeat exactly after a certain time period and streamwise length; in addition, these orbits may also have azimuthal rotations
A dynamical systems approach has also been pursued in studies of the laminarturbulent transition [
21]; as mentioned, TWs have been identified for
$R{e}_{D}$ lower than the observed transition. It has also been shown that spatially localised RPOs can experience a series of bifurcations leading to transient chaos.
3. Magneticallybounded turbulent flow
For the material on PP, we focus on commonalities with FM, therefore many specific features have been disregarded. Of course this entails a risk of leaving out important topics. An example of what is left out is specific issues relating to electromagnetic (EM) fields and plasma currents.
A note on units: In PP, temperature is usually stated using units of energy, where 1 eV corresponds to around 11 600 K. Another convention to keep in mind is that for PP, density has the units of particle density (number of particles per volume), whereas in FM, mass density is used (mass per volume).
3.1. Magnetic field structure
A plasma consists of charged particles (electrons and ions), which need to be confined within a toroidal shape to enable fusion. Since charged particles follow magnetic field lines (with superimposed gyroradii), the method of confinement is to construct closed magnetic field surfaces.
The basic shape of a magnetic confinement devices is a torus, with coordinates (i) toroidal (the "long" way around a torus), (ii) radial and (iii) poloidal (the "short" way around a torus).
Relating to pipe flow, the corresponding coordinates are toroidal/streamwise, radial/wallnormal and poloidal/spanwise.
Additional (a) perpendicular and (b) parallel coordinates refer to the directions perpendicular (crossfield) and parallel to the magnetic field. These are different from  but related to  the toroidal, radial and poloidal coordinates.
We focus on cases from tokamaks [
73] but include material on stellarators and heliotrons [
74] when relevant.
For these machine types, the main toroidal magnetic field is generated by external planar coils. A main difference between tokamaks and stellarators/heliotrons is how the poloidal magnetic field is created: In tokamaks, it is created by a toroidal current induced through transformer action, but in stellarators/heliotrons it is created by modular (nonplanar) coils. For stellarators, the modular coils are predominantly poloidal whereas for heliotrons, the modular coils are mainly toroidal. This implies that the plasma current in tokamaks is much higher than in stellarators/heliotrons, which has important implications for e.g. currentdriven instabilities, steadystate operation and machine complexity.
All machine types treated herein generate an MHD equilibrium with nested magnetic surfaces. The boundary is named the last closed flux surface (LCFS) which is called a separatrix if it includes one or more "Xpoints", which are points with zero (null) poloidal field. Plasmas can also be bound by physical limiters. We use the term "magneticallybounded" for plasmas which are bounded by a separatrix, i.e. where the LCFS is not in contact with physical surfaces. The region between the separatrix and the physical wall is called the scrapeoff layer (SOL), where magnetic field lines are open and intersect the wall. Divertors intersect the open field lines from the separatrix and are used for particle and heat exhaust.
The winding number of the magnetic field lines is called the safety factor in tokamaks due to its importance for plasma stability:
where
$\varphi $ is the toroidal angle and
$\theta $ is the poloidal angle. Traditionally, another definition has been used in stellarators/heliotrons:
Typically,
qprofiles in tokamaks have a minimum
${q}_{\mathrm{min}}$ close to the axis and increase towards the plasma edge. For stellarators/heliotrons, the
$\iota $profile is often more flat. We define the magnetic shear:
where
r is the minor radius measured from the magnetic axis.
Thus, tokamaks have high shear and stellarators/heliotrons have low shear.
The magnetic field decreases from the center of the torus outwards inversely proportional to the major radius R, which can lead to particle trapping due to the magnetic mirror effect. For tokamaks, these are called banana orbits and centered on the outboard midplane (the low field side). For stellarators/heliotrons, the particles are helically trapped.
3.2. Turbulence and improved confinement regimes
As mentioned, the purpose of the magnetic field is confinement; the plasma also needs to have a sufficiently high temperature for the ions to fuse and release energy. Two timescales can be used to quantify energy and particle confinement, namely the energy confinement time ${\tau}_{E}$ and the particle confinement time ${\tau}_{p}$. These timescales indicate how efficient the confinement of energy (temperature) and particles (density) is.
Another way of gauging confinement quality is $\beta $, which is the plasma pressure normalised to the magnetic pressure. $\beta $ can be defined both using the total (B), the toroidal (${B}_{\varphi}$) or the poloidal (${B}_{\theta}$) magnetic field. As the plasma pressure increases, the center of the magnetic axis is displaced radially outwards, an effect called the Shafranov shift.
If transport is only taking place due to thermal motion (Coulomb collisions), with curvature effects included, it is called neoclassical transport [
75]. However, in reality much larger transport is observed perpendicular to the magnetic field, which is called anomalous transport [
76].
Anomalous transport is caused by turbulence, e.g. microinstabilities driven by the ion (ITG) or electron (ETG) temperature gradient or by trapped electrons such as the trapped electron mode (TEM). The smallest turbulent scale is due to ETG, medium scale due to TEM and largest scale due to ITG. Instabilities driven by density or temperature gradients are called drift waves (DW). Even larger scale (macroscopic) MHD instabilities can be driven by e.g. current, pressure or fast particles. Often instabilities can be ballooning, which means that  due to curvature effects  their growth rate is larger on the outer side of the torus compared to the inner side. Turbulence can lead to the formation of streamers, first identified for ETG turbulence [
77,
78] followed by ITG turbulence [
79]. Streamers are radially elongated mesoscale vortices centered on the outboard midplane; they lead to enhanced crossfield transport, thereby degrading confinement.
A main effort in the fusion community is to understand and reduce anomalous transport to improve confinement and obtain more efficient fusion reactions.
One way to control anomalous transport is by external heating of electrons and ions, for example by ion or electron cyclotron resonance heating (ICRH/ECRH) or by neutral beam injection (NBI). The plasma current can also be manipulated both using external heating and current drive, e.g. lower hybrid current drive (LHCD).
The plasma state can experience either gradual confinement improvements or sudden bifurcations to improved confinement regimes; sometimes improved confinement is associated with instabilities such as edge localised modes (ELMs), which lead to bursts of crossfield transport of particles and energy. Other improved confinement regimes can be associated with coherent modes which regulate transport and avoid ELMs.
3.3. Length scales
An important group of length scales is associated with the Larmor radius, which is the gyration distance of charged particles around the magnetic field:
where the subscript
j represents electrons (
e) or ions (
i),
${m}_{j}$ is the mass,
${v}_{\perp}$ is the velocity perpendicular to the magnetic field,
${e}_{j}$ is the charge and
${\omega}_{cj}={e}_{j}B/{m}_{j}$ is the cyclotron frequency. Here, we can relate the velocity to temperature by assuming two degrees of freedom:
which leads to:
For scaling purposes, the ion Larmor radius normalised to the minor radius of the machine (
$r=a$) is used:
and for turbulence modelling, the ion Larmor radius at the electron temperature is used:
Scale lengths have been mentioned previously in
Section 2.9; we generalise the notation to write the scale length
${L}_{x}$ of a quantity
x as:
Equation (
9) can be reformulated for electron density fluctuations (
$x={n}_{e}$):
where
$\delta {n}_{e}$ are density fluctuations (corresponding to the friction velocity) and
${\rho}_{s}$ is the typical scale of the density fluctuations (corresponding to the mixing length). For DWs, the density fluctuations saturate at this level:
where
${k}_{\perp}\sim 1/{\rho}_{s}$ is the perpendicular wavenumber of the density fluctuations.
Microscales are on the order of the (ion/electron) Larmor radius, from submm to mm scales, depending on temperature and magnetic field strength. Macroscales are on the order of the machine minor radius and mesoscales are between micro and macroscales; an example of a mesoscale phenomenon is streamers, and we will encounter other mesoscale structures later.
An effect known as turbulence spreading occurs for inhomogeneous turbulence [
80]: "Turbulence spreading is a process of turbulence selfscattering by which locally excited turbulence spreads from the place of excitation to other places." This is not related to the K41 paradigm which deals with homogeneous turbulence.
3.4. Rational safety factors and transport
If $q=m/n$ is a rational number (m and n both integers), then the magnetic field line returns to the initial position after m toroidal and n poloidal rotations. For a fixed toroidal angle, this corresponds to a poloidal mode number m and for a fixed poloidal angle it corresponds to a toroidal mode number n.
Since the magnetic field line paths constitute a Hamiltonian system, rational values of the safety factor correspond to resonant tori, which are unstable against perturbations [
16]. Perturbations can lead to the formation of magnetic islands or ergodic regions.
A classical example of instabilities is sawtooth crashes (relaxations) for
$q<1$, where heat and particles are ejected from the core plasma due to magnetic reconnection [
81]: "Magnetic reconnection is a topological rearrangement of magnetic field that converts magnetic energy to plasma energy." The periodic core temperature collapse is due to an instability which has an
$m=n=1$ structure, corresponding to
$q=1$.
Enhanced transport has been observed for
qprofiles at or close to loworder rationals in the Rijnhuizen Tokamak Project (RTP) [
82,
83]. Transport barriers for the electron temperature were observed as temperature steps which could be controlled by the deposition location of external electron heating. A "qcomb" model was constructed to model the transport barriers as low electron heat conductivity at loworder rationals, possibly due to the formation of magnetic island chains.
As for the RTP tokamak, a similar behaviour has been observed in the Wendelstein 7Advanced Stellarator (W7AS) [
84,
85]. Here, reduced transport was also found to be associated with loworder rationals.
3.5. Magnetic islands caused by instabilities or topology
In both tokamaks and stellarators/heliotrons, magnetic islands can be caused by instabilities as mentioned above, e.g. global Alfvén eigenmodes (GAE) [
86]. These islands can be either nonrotating ("locked") or rotating.
In addition, natural magnetic islands can exist in stellarators/heliotrons. An example is from the W7AS and Wendelstein 7X (W7X) stellarators, where islands form for
where the constant "5" is due to the fact that the machines have a fivefold toroidal symmetry. The five field periods are also flip symmetric, leading to ten identical sections. For W7AS the standard divertor configuration (SDC) was
$m=9$ [
87] whereas for W7X it is
$m=5$ [
88], the change being due to
$\iota $profile differences. Thus, W7X has larger islands with lower poloidal mode numbers compared to W7AS.
The natural magnetic islands can be used to form a separatrix and an associated island divertor. This also enables detachment, which is a state where a large fraction of the power is dissipated by volume radiation before it reaches the physical wall. This is a potential exhaust solution under reactor conditions, since the heat flow will be intercepted before reaching the divertor target plates, leading to significantly reduced fluxes at the targets.
3.6. $E\times B$ flow shear decorrelation
A mechanism to reduce turbulent transport by velocity shear has been identified in [
89] and reviewed in [
10], with earlier efforts in e.g. [
90]. It causes eddy stretching which leads to eddies losing coherence (breakup), i.e. energy transfer from large scales (low wavenumbers) to small scales (high wavenumbers). It is called sheared
$E\times B$ flow and is generated by the radial electric field
${E}_{r}$ which results from the radial force balance (ignoring the RS term):
where the "
i" subscript refers to ions (dominating compared to electrons),
p is the pressure,
Z is the charge state,
e is the electronic charge,
${v}_{\varphi}$ is the toroidal velocity and
${v}_{\theta}$ is the poloidal velocity. Suppression of turbulence takes place if the shearing rate
${\omega}_{E\times B}$ is larger than the maximum linear growth rate
${\gamma}_{\mathrm{max}}$ of the relevant instability:
The shearing rate increases with shear in the radial electric field
$\partial {E}_{r}/\partial r$, so the regions where the radial electric field changes rapidly as a function of radius are the regions where turbulence is suppressed most efficiently.
$E\times B$ shearing is a mean flow effect on turbulence which affects not only the turbulence amplitude, but also the "phase angle between an advected fluctuation and the advecting flow" [
10]. Shear suppression is a universal, selfregulating process between shear flow and transport: Turbulence reduction leads to steepened gradients (temperature, density), which increases the pressure gradient, which in turn increases the flow shear and reduces turbulence further.
In addition to the shearing rate criterion, three additional requirements have to be fulfilled:
These requirements are often met in fusion plasmas, but rarely in nonionised fluids; some exceptions are mentioned in [
10], e.g. stratospheric geostrophic flow and perhaps the laminar phase between bursts of turbulence for wallbounded flows.
3.7. Transport barriers
In this section we provide a brief overview of the different TB variants in fusion plasmas: (i) ETB [
4,
91], (ii) ITB [
6,
7,
92,
93] and (iii) both ETB and ITB [
94].
3.7.1. ETB
As mentioned in the Introduction, the first ETB was identified in 1982 in the Axially Symmetric Divertor Experiment (ASDEX) tokamak [
4]. For NBI power above a certain threshold, an LHmode transition was obtained. This was possible for diverted plasmas but not for limited plasmas. Apart from the power threshold, Hmode could only be accessed for a safety factor at the edge
${q}_{a}>2.6$.
The improved Hmode confinement was seen as an increased poloidal $\beta $ (${\beta}_{p}$) and an increase of the electron density and temperature. Bursts of ${H}_{\alpha}{D}_{\alpha}$ emission were observed in Hmode which were later identified as ELM signatures.
The Hmode ETB is quite robust and has steep density and temperature gradients just inside the LCFS. $E\times B$ flow shear is part of the prerequisite for the Hmode, along with suitable edge plasma conditions which may vary between different machine designs. As of now, there is no comprehensive, predictive theorybased model for ETB formation and spatial structure.
ELMs generated by the large pressure gradients created in ETBs can often degrade or even destroy the barrier. Some methods exist to stabilise instabilities, for example applying an external magnetic field or operating variants of Hmodes with quasicoherent (QC) or edge harmonic oscillations (EHO), which provide increased particle transport through barrier withhout significantly increasing the energy transport.
3.7.2. ITB
As referred to in the Introduction, the first ITBs were identified in 1995 in two tokamaks, the Tokamak Fusion Test Reactor (TFTR) [
6] and the Doublet IIID (DIIID) [
7].
For both machines, the most important component to achieve an ITB was to get reversed magnetic shear which was obtained by creating a hollow current density profile. This was done by a combination of current ramping and NBI and took advantage of the fact that the current diffusion time is much longer than the rise time of the plasma current.
The ITB led to reduced particle and ion thermal transport in the plasma core where reversed shear was created. The high pressure gradient generated strong offaxis bootstrap current which helped to maintain the hollow current density profile. Electron thermal transport was also reduced but not as significantly as the ion thermal transport. The ion thermal diffusivity and electron particle diffusivity decrease to close to or below the neoclassical level.
MHD modes can exist outside the ITB and act to limit the obtainable $\beta $.
ITBs in tokamaks were reviewed in [
92]. It was found that low or reversed magnetic shear in combination with large
$E\times B$ shear flows are essential ITB ingredients, where magnetic shear stabilises high
n ballooning modes and
$E\times B$ shear stabilises medium to longwavelength turbulence, i.e. ion thermal transport and particle transport. It is possible to have high electron thermal transport even with ITBs. The
q value at 95% of the magnetic flux,
${q}_{95}$, is found to be important for magnetic stability and
${q}_{\mathrm{min}}$ has been seen to correlate with the ITB foot. The Shafranov shift can have a stabilising effect on turbulence called
$\alpha $stabilisation. ITBs can exist with equal ion (
${T}_{i}$) and electron (
${T}_{e}$) temperatures, but also for cases where
${T}_{i}<{T}_{e}$ or
${T}_{e}<{T}_{i}$, depending on the plasma density and external heating method.
In the rest of the section we summarise results from the most recent review [
93] which covers both tokamak and helical (in our case: Stellarators/heliotrons) plasmas. A systematic approach is applied, with an ITB definition being a (radial) discontinuity of temperature, flow velocity or density gradient.
ITBs are characterised by three parameters:
Normalised temperature gradient $R/{L}_{T}=R\times \nabla T/T$ (large value: weak, small value: strong)
Location ${r}_{\mathrm{ITB}}/a={\rho}_{\mathrm{ITB}}=({\rho}_{\mathrm{shoulder}}+{\rho}_{\mathrm{foot}})/2$ (large value: large, small value: small)
Width $W/a={\rho}_{\mathrm{foot}}{\rho}_{\mathrm{shoulder}}$ (large value: wide, small value: narrow)
Here, ${L}_{T}=T/\nabla T$ is the temperature scale length, "shoulder" is at the top of the steep gradient and "foot" is at the bottom of the steep gradient.
The key elements for ITB formation are summarised as:
It is instructive to write the equations relating radial fluxes (particle, momentum, electron/ion heat) and gradients (density, toroidal rotation, temperature). For the particle flux
$\Gamma $ we write:
where
D is the diffusion coefficient,
${n}_{e}$ is the electron density and
${v}_{\mathrm{conv}}$ is the convection velocity. For the momentum flux
${P}_{\varphi}$ we write:
where
${m}_{i}$ is the ion mass,
${\nu}_{\perp}$ is the perpendicular kinematic viscosity,
${v}_{\mathrm{pinch}}$ is the momentum pinch velocity and
${\Gamma}_{\varphi}^{\mathrm{resi}}$ is the radial flux due to residual stress [
95]. We note that Equation (
28)  when disregarding the two final righthand side terms  has the same structure as Equation (
4). For the electron and ion heat flux (
${Q}_{e,i}$) we write:
with the electron and ion thermal diffusivity
${\chi}_{e,i}$. To cite [
93]: "When the density, velocity, and temperature gradient become large due to the decrease in the diffusion coefficient,
D, viscosity,
${\mu}_{\perp}$, and thermal diffusivity,
${\chi}_{e,i}$, the region in the plasma is called the transport barrier." We will use diffusion coefficient as a collective term for
D,
${\mu}_{\perp}$ and
${\chi}_{e,i}$. An ITB can be defined as a bifurcation in the fluxgradient relationship, which causes a discontinuity in the density/velocity/temperature gradient for a given particle/momentum/heat flux, leading to the formation of a discontinuity in the gradient with radius.
The ITB foot (point) often follows integer q values, typically $q=1$ ($\rho \sim 0.3$), $q=2$ ($\rho \sim 0.5$) and $q=3$ ($\rho \sim 0.7$); this is valid for positive or weakly reversed magnetic shear, but not strong reversed magnetic shear. Sometimes ITBs are also observed for halfinteger q values. For reversed magnetic shear, an ITB appears when ${q}_{\mathrm{min}}$ crosses a rational surface.
Experiments using resonant magnetic perturbations (RMPs) to produce magnetic islands were carried out in the Large Helical Device (LHD) to distinguish the role of magnetic islands and rational surfaces. It was found that [
93]: "This experiment supports the idea that the magnetic island at the rational surface contributes to the transition from the Lmode to the ITB rather than to the rational surface itself." A reduction of transport inside magnetic islands has been observed, close to what is called the "Opoint", as opposed to the previously mentioned Xpoint. There is a reduction in turbulence (and transport) at the boundary of magnetic islands and the pressure profile is flat in the Opoint inside the islands.
Important differences between tokamak and helical plasmas include:
Ion barriers are most significant for tokamaks, electron barriers for helical devices
Simultaneous ion/electron barriers have been seen in tokamaks, but not in helical devices
In general, magnetic shear is negative for helical devices, but both positive and negative for tokamaks
Differences in particle transport: Clear density barrier for tokamaks, barrier disappears for higher density in helical devices. But it exists for both when pellet injection is used.
The toroidal angular velocity is higher for tokamaks
The sign of the impurity pinch is opposite: Inwards for tokamaks (impurity accumulation), outwards for helical systems
ITBs are more variable for tokamaks due to the freedom of the current profile (magnetic shear), which is restricted in helical devices

Radial electric field:
Nonlocality of ITB plasmas has been observed, e.g. coupling between the inside and the outside of the ITB. The curvature of the ion temperature (${\partial}^{2}{T}_{i}/\partial {r}^{2}$) has been linked with ITB stability, where a convex (concave) curvature means a less (more) stable ITB, respectively.
3.7.3. Both ETB and ITB
It was already demonstrated in [
7] that an ITB can coexist with both L and Hmode edges, where an ITB with an Hmode edge is a double barrier (DB), i.e. an ETB and an ITB. Nonlocality has also been observed for this type of DB, where the ITB formation takes place simultaneously with the LH transition [
93].
Multiple barriers have been reviewed in [
94] and we present a summary of this work in the rest of the section.
The leading mechanisms for stabilisation are stated as (i) $E\times B$ flow shear and (ii) reduction of growth rates due to $\alpha $stabilisation. The combination of ETB and ITB is useful if it can:
Increase the plasma volume with reduced transport
Lead to improved stability against MHD modes
For tokamaks: Improve the bootstrap current fraction for steadystate operation
On the other hand, potential drawbacks include:
ITB degradation due to ETB, e.g. reduction of rotation shear and pressure gradient at the ITB location
High density at the ETB can reduce NBI penetration efficiency
ELMs can lead to flattening of ITB temperature gradients
An example where the barriers lead to additive beneficial effects is the quiescent double barrier (QDB) mode in DIIID, where an ITB is combined with a quiescent Hmode (QH) which has an EHO.
3.8. Zonal flows
ZFs are azimuthally symmetric bandlike $E\times B$ shear flows with mode numbers $n=m=0$. They are mesoscale electric field fluctuations with zero mean frequency and finite radial wave number ${k}_{r}$. ZFs are flows which are driven by turbulence, e.g. turbulent shear RS or DW. Due to their structure, ZFs are benign repositories for free energy and do not drive radial (energy or particle) transport. ZFs vary rapidly in the radial direction. For toroidal plasmas having a strong toroidal magnetic field (valid assumption in this review), ZFs are predominately poloidally directed with velocities ${v}_{\theta}={E}_{r}/B$ and ${v}_{\varphi}=2q{v}_{\theta}cos\theta $. The convention is that $\theta ={0}^{\circ}$ at the outboard midplane and increases in the counterclockwise direction.
ZFs differ from mean
$E\times B$ shear flows (see
Section 3.6); mean shear flows are generated as a result of the ion radial force balance and ZF shear flows are driven by turbulence. Mean shear flows can persist without turbulence, whereas ZF shear flows cannot. This is reflected in the different radial electric fields:
The radial electric field from ZFs is oscillatory, complex, consists of small structures and is driven exclusively by nonlinear wave interaction processes.
The mean radial electric field evolves on transport timescales and is driven by e.g. heating, fuelling and momentum input which determine equilibrium profiles, in turn regulating the radial force balance.
The mean and ZF shear flows can interact, e.g. mean flows can suppress ZFs through turbulence decorrelation. Both flow types can tilt and break turbulent eddies.
ZFs shear or quench turbulence to extract energy from it leading to a selfregulating mechanism with a predatorprey system of turbulent energy (prey) and ZF energy (predator). In that sense, ZFs can shift (delay) the onset of turbulence, often referred to as the "Dimits shift" [
102].
ZFs have been linked to rational $\iota $ values, e.g. in the H1 National Facility (H1NF), which was a 3field period heliac. ZFs were found at two locations (due to reversed shear) where $\iota =7/5$.
Because of the 3D nature of shear flow physics, several RS terms can contribute to ZF generation, e.g. radialparallel, radialperpendicular and radialpoloidal.
ZFs are not Landau (wave) damped but mainly collisionally damped due to friction between trapped and circulating ions; they increase with decreasing collisionality.
The energy partition between ZFs and turbulence is key for plasma confinement: A large fraction of ZFs results in better confinement. To understand the process, one can write the ratio of ZFs and turbulence as:
where
V is the ZF intensity,
N is the turbulence energy,
${\gamma}_{L}$ is the DW (turbulence) linear growth rate,
$\alpha $ is a coupling constant between ZFs and DWs and
${\gamma}_{\mathrm{damp}}$ is the flow damping of ZFs due to collisionality. The ratio
$\zeta $ increases with improved confinement since the damping rate decreases.
ZFs may take the role of a trigger for confinement transitions, possibly at the LH transition. An interaction between mean and zonal flows may also exist; e.g that the mean $E\times B$ flow exists before the transition and that the additional effect of ZFs triggers the transition itself.
In nature, the Jovian belts/zones and the terrestrial jet stream have been given as examples of ZFs.
Finally we mention zonal fields which is the generation of structured magnetic fields from turbulence, i.e. a magnetic counterpart to ZFs. The magnetic field structures, also with $n=m=0$ and finite radial wave number, can be generated by DW turbulence and may have a backreaction on turbulence via magnetic shearing.
3.9. Geodesic acoustic modes
We proceed with a review of geodesic acoustic modes (GAMs) based on material in [
97,
98,
99,
101].
In many respects, GAMs are similar to ZFs: GAMs also have mode numbers $n=m=0$, but couple to pressure/density fluctuations with $m=\pm 1$ (poloidal mode number) and $n=0$. These fluctuations are poloidally asymmetric and highest at the top and bottom of tokamak plasmas. For stellarators/heliotrons, the highest fluctuation is not at the top and bottom, but follows the helical pitch. For completeness, we note that there is also a magnetic component with $m=\pm 2$ and $n=0$. GAMs have velocities ${v}_{\theta}={E}_{r}/B$ and ${v}_{\varphi}={q}^{1}{v}_{\theta}cos\theta $.
GAMs have a finite frequency as opposed to ZFs which have zero frequency. The GAM frequency scales with the square root of the temperature; this can be derived from singlefluid ideal MHD:
where:
is the speed of sound and
$\gamma =5/3$ is the specific heat ratio.
GAMs are both Landau damped ($\propto exp({q}^{2})$) and collisionally damped; the zero frequency ZFs are not Landau damped, but only collisionally damped. Due to the differences in Landau damping and magnetic configuration in tokamaks and helical devices, GAMs are mainly found at the edge of tokamaks and in the low $\iota $ core region of stellarators/heliotrons. Generally, it has also been observed that GAMs are stronger (and have been observed more often) in tokamaks than helical devices.
GAMs can be driven directly from the poloidally symmetric
$m=0$ component of the turbulent shear RS (
${\nabla}_{r}\langle {\tilde{v}}_{r}{\tilde{v}}_{\theta}\rangle $), similar to ZFs: "Since both the GAM and the ZF are driven by turbulence there is the issue of competition in the nonlinear transfer leading to the dominance of one or other mode." [
101]. However, ZFs and GAMs can coexist and transitions between ZFs and GAMs have also been observed.
Both ZFs and GAMs have comparable radial correlation lengths, which are mesoscale as found for streamers as well.
The response of ZFs and GAMs to fluctuations is different: ZFs are incompressible (slow response) and GAMs are compressible (fast response).
Usually, GAMs are not observed in Hmode.
The impact of GAMs on transport can be summarised as:
No direct radial energy or particle transport
Oscillatory flow shearing
Act as an energy sink through Landau damping or dissipation
Modulate crossfield transport through pressure fluctuations (GAMs are rarely contiguous and stable)
Finally, we collect quotes from [
101] on the relationship between GAMs and magnetic islands:
"The interaction of GAMs with MHD modes (static and rotating) is multifold. An island chain may create a GAMlike oscillation, or it may enhance and/or entrain a natural edge GAM, or it may suppress and destroy the natural GAM."
"At the extreme, the velocity shearing associated with the GAM can also restrict the island radial structure and thus limit the growth of the MHD mode."
"The flow and turbulence behaviour can be divided into three distinct spatial regions: inside the island separatrix, around the island boundary, and spatially (radially) well away from the island chain."
3.10. Blobs
Blobs are filaments generated by edge plasma turbulence with enhanced levels of particles and heat aligned along magnetic field lines in the SOL [
103]. There is intermittent eruptions of plasma and heat into the SOL which leads to radial motion of blobs. They are ballooning, with more transport at the outboard midplane. The fluctuation level and turbulencedriven transport (number of events) increases with
$\beta $ and collisionality. Blobs have an asymmetric waveform with time, where the rise time is fast and the decay is slow; their total duration is of order 25 ms.
A theory on blob creation based on breakup of streamers due to velocity shear has been experimentally validated in [
104]. These streamers are located outside the separatrix, so in that sense they are different from the streamers previously mentioned. A possible mechanism for the shear flow generation is the interchange instability, which is "very similar in nature to the RayleighTaylor instability in fluid dynamics" [
74]. More blobs are observed in Lmode than in Hmode.
8. Conclusions
We have presented a comparative study of wall and magneticallybounded turbulent flows to identify possible crossdisciplinary similarities. The most important common phenomena found are transport barriers, coherent (turbulent) structures and shear Reynolds stress flow generation.
To the best of our knowledge, this is first time the uniform momentum zones in fluid mechanics have been compared to internal transport barriers in magnetically confined fusion plasmas.
Exact coherent structures found in fluid mechanics appear to have many similarities with magnetic islands in fusion plasmas which are associated with rational values of the winding number of the magnetic field lines.
Zonal flows in fusion plasmas create radial velocity shear which is also seen between uniform momentum zones in nonionised turbulent flows.
We propose that these phenomena are common (universal) ingredients for both nonionised fluids and magnetically confined fusion plasmas:
Uniform zones (momentum, heat, concentration)/Internal transport barriers
Exact coherent states/Magnetic islands
Shear Reynolds stress driven (zonal) flows
The improved understanding has been used to reinterpret transport barriers and core turbulence.
An additional potential similarity is between rampcliff structures in passive scalar flows and sawtooth crashes caused by magnetic reconnection in fusion plasmas.
Finally, we propose a new crossdisciplinary experimentallybased research program to test the ideas we have put forth.
A note of caution: Crossdisciplinary research is notoriously difficult both to carry out and to gauge, since you will be an outsider in some fields and risk being seen as a crackpot in others. This naturally leads to the disclaimer that all misunderstandings and errors are mine.