4.1. Perpendicular kinetic energy equations

This section will focus on the kinetic energy in the perpendicular velocity fluctuations and its relation to the other energy forms in the plasma. We define the perpendicular total ($E_{{k},\perp }$), mean-flow ($E_{k,m,\perp }$) and turbulent ($k_{\perp }$) kinetic energies as

(4.1a–c)\begin{equation} E_{{k},\perp} \triangleq \frac{m\boldsymbol{V}_{0,\perp}^2}{2},\quad E_{k,m,\perp} \triangleq \frac{m\tilde{\boldsymbol{V}}_{0,\perp}^2}{2},\quad \bar{n} k_{{\perp}} \triangleq \frac{m\overline{n \boldsymbol{V}_{0,\perp}''^2}}{2}, \end{equation}

where $\boldsymbol {V}_{0,\perp }$ contains the perpendicular velocity components that are relevant for the ion inertia. As discussed in the previous section, we assume that $\boldsymbol {V}_{0,\perp }=\boldsymbol {V}_{E}+\boldsymbol {V}_{*,i}$. Note that $E_{{k},\perp }$ varies rapidly in time and space as it follows the instantaneous fluctuations, while $E_{k,m,\perp }$ and $k_{\perp }$ are ensemble-averaged quantities that do not change at these small scales. The latter two are constant in time in a statistical steady state, while the former is not. Note also that the sum of mean flow and turbulent kinetic energy per unit volume equals the averaged total kinetic energy per unit volume

(4.2)\begin{equation} \overline{nE_{{k},\perp}}=\bar{n} E_{k,m,\perp} + \bar{n} k_{{\perp}}. \end{equation}

The derivation of the turbulent kinetic energy equation presented here is based on the procedure applied by Scott (Reference Scott2003). Garcia et al. (Reference Garcia, Naulin, Nielsen and Rasmussen2006) and Tran et al. (Reference Tran, Kim, Jhang and Kim2019) likewise followed a similar scheme. These derivations are refined here by taking the density fluctuations into account in the averaging operators and by considering a number of terms which were neglected in these references. To limit the number of closure terms, we make use of the Favre averages introduced in § 3. Note that an analogous procedure has also been applied to a reduced turbulence description for a 2-D, electrostatic, sheath-limited, $E\times B$-only SOL in earlier work (Coosemans et al. Reference Coosemans, Dekeyser and Baelmans2021b).

The total perpendicular kinetic energy equation is obtained by multiplying the charge balance equation (2.20) with the electrostatic potential

(4.3)\begin{equation} \phi\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{J}_{p} =-\phi\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{J}_{\|} - \phi\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{J}_{*} ,\end{equation}

and rewriting the different terms. The polarisation current on the left-hand side introduces the kinetic energy in the equation. This term is first rewritten as

(4.4)\begin{equation} \phi\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{J}_{p} = \boldsymbol{\nabla}\boldsymbol{\cdot}\phi\boldsymbol{J}_{p} - \left(\boldsymbol{\nabla}_{{\perp}}\phi+\frac{\boldsymbol{\nabla}_{\perp} p_{{i}}}{en}\right)\boldsymbol{\cdot}\boldsymbol{J}_{p} + \boldsymbol{\nabla}_{\perp} p_{{i}}\boldsymbol{\cdot}\boldsymbol{V}_{p}. \end{equation}

In this expression, the second term on the right-hand side is in turn rewritten as

(4.5)\begin{align} -\left(\boldsymbol{\nabla}_{{\perp}}\phi + \frac{\boldsymbol{\nabla}_{\perp} p_{{i}}}{en}\right)\boldsymbol{\cdot}\boldsymbol{J}_{p} & = \frac{\partial nE_{{k},\perp}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}(nE_{{k},\perp}\boldsymbol{V}_{C} + \boldsymbol{\varPi}\boldsymbol{\cdot}\boldsymbol{V}_{0,\perp} ) \nonumber\\ & \quad - \boldsymbol{\varPi}:\boldsymbol{\nabla}\boldsymbol{V}_{0,\perp}^{\rm T} + mnV_{\|}\frac{{\rm D}\boldsymbol{b}}{{\rm D} t} \boldsymbol{\cdot}\boldsymbol{V}_{0,\perp} + E_{{k},\perp} S_{n,i} -\boldsymbol{S}_{m}\boldsymbol{\cdot}\boldsymbol{V}_{0,\perp}. \end{align}

In this derivation, we made use of continuity equation (2.1) and the definition of the polarisation current (2.9) (supplemented with (2.11)), with the convection velocity given by $\boldsymbol {V}_{C}$ in both equations. Furthermore, common vector calculus identities were used.

The first term on the right-hand side of (4.3) is straightforwardly rewritten as $\phi \boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {J}_{\|}=\boldsymbol {\nabla }\boldsymbol {\cdot }\phi \boldsymbol {J}_{\|}-J_{\|}\boldsymbol {\nabla }_{\|}\phi$. Following Scott (Reference Scott2003), the second term is then rewritten as

(4.6)\begin{align} -\phi\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{J}_{*} & =-\boldsymbol{\nabla}\boldsymbol{\cdot}\phi\boldsymbol{J}_{*} + \boldsymbol{\nabla}\phi\boldsymbol{\cdot}\boldsymbol{J}_{*} =-\boldsymbol{\nabla}\boldsymbol{\cdot}\phi\boldsymbol{J}_{*} - \boldsymbol{\nabla} p\boldsymbol{\cdot}\boldsymbol{V}_{E} \nonumber\\ & =-\boldsymbol{\nabla}\boldsymbol{\cdot}(\phi\boldsymbol{J}_{*} + p\boldsymbol{V}_{E}) + p\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{V}_{E}. \end{align}

Filling out expressions (4.4)–(4.6) in (4.3), the total perpendicular kinetic energy equation is found

(4.7)\begin{align} & \frac{\partial nE_{{k},\perp}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}(nE_{{k},\perp}\boldsymbol{V}_{C} + \boldsymbol{\varPi}\boldsymbol{\cdot}\boldsymbol{V}_{0,\perp} + \phi\boldsymbol{J} + p\boldsymbol{V}_{E}) = J_{\|}\nabla_{\|}\phi +p\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{V}_{E} \nonumber\\ & \quad + \boldsymbol{\varPi}:\boldsymbol{\nabla}\boldsymbol{V}_{0,\perp}^{\rm T} - \boldsymbol{\nabla}_{\perp} p_{{i}}\boldsymbol{\cdot}\boldsymbol{V}_{p} - mnV_{\|}\frac{{\rm D}\boldsymbol{b}}{{\rm D} t}\boldsymbol{\cdot}\boldsymbol{V}_{0,\perp} -E_{{k},\perp} S_{n_{i}} + \boldsymbol{S}_{m}\boldsymbol{\cdot}\boldsymbol{V}_{0,\perp}. \end{align}

To arrive at equations for $E_{k,m,\perp }$ and $k_{\perp }$ defined in (4.1a–c), the $E_{{k},\perp }$ equation (4.7) should be averaged and split in a contribution due to mean flows and a contribution due to fluctuations. The procedure to obtain an equation for $E_{k,m,\perp }$ is rather similar to that used for $E_{{k},\perp }$. For this, we start from the averaged charge balance equation multiplied with the average electrostatic potential

(4.8)\begin{equation} \bar{\phi}\boldsymbol{\nabla}\boldsymbol{\cdot}\bar{\boldsymbol{J}}_{p} =-\bar{\phi}\boldsymbol{\nabla}\boldsymbol{\cdot}\bar{\boldsymbol{J}}_{\|} - \bar{\phi}\boldsymbol{\nabla}\boldsymbol{\cdot}\bar{\boldsymbol{J}}_{*}. \end{equation}

The terms on the right-hand side will be manipulated in exact analogy to those in the total kinetic energy equation. Some slight complications arise for the polarisation current term, however, because we need the dot product between $\bar {\boldsymbol {J}}_{p}$ and the Favre-averaged potential gradient to form the time rate of change of the mean-field perpendicular kinetic energy $E_{k,m,\perp }$

(4.9)\begin{equation} \bar{\phi}\boldsymbol{\nabla}\boldsymbol{\cdot}\bar{\boldsymbol{J}}_{p} = \boldsymbol{\nabla}\boldsymbol{\cdot}\bar{\phi}\bar{\boldsymbol{J}}_{p} - \left(\widetilde{\boldsymbol{\nabla}_{{\perp}}\phi} + \frac{\boldsymbol{\nabla}_{{\perp}}\bar{p}_i}{e\bar{n}}\right)\boldsymbol{\cdot}\bar{\boldsymbol{J}}_{p} + \boldsymbol{\nabla}_{{\perp}}\bar{p}_i\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{p} + \frac{\bar{\boldsymbol{J}}_{p}}{\bar{n}}\boldsymbol{\cdot}\overline{n'\boldsymbol{\nabla}\phi'}. \end{equation}

The last term on the right-hand side, which we will call ‘Favre averaging term’, arises because Favre averages and gradients do not commute. This then leads to the following equation:

(4.10)\begin{align} & \frac{\partial \bar{n}E_{k,m,\perp}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}(\bar{n}E_{k,m,\perp}\tilde{\boldsymbol{V}}_{C} + m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_0''}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{0,\perp} + \bar{\boldsymbol{\varPi}}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{0,\perp} + \bar{\phi}\bar{\boldsymbol{J}} + \bar{p}\bar{\boldsymbol{V}}_{E}) \nonumber\\ & \quad =\bar{J}_{\|}\nabla_{\|}\bar{\phi} + \bar{p}\boldsymbol{\nabla}\boldsymbol{\cdot}\bar{\boldsymbol{V}}_{E} + m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{0,\perp}''}: \boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{0,\perp}^{\rm T} + \bar{\boldsymbol{\varPi}}:\boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{0,\perp}^{\rm T} - \boldsymbol{\nabla}_{{\perp}}\bar{p}_i\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{p} \nonumber\\ & \qquad - m\overline{nV_{\|}\frac{{\rm D}\boldsymbol{b}}{{\rm D} t}}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{0,\perp} - \frac{\bar{\boldsymbol{J}}_{p}}{\bar{n}}\boldsymbol{\cdot}\overline{n'\boldsymbol{\nabla}\phi'} - E_{k,m,\perp} \bar{S}_{n_{i}} + \bar{\boldsymbol{S}}_{m}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{0,\perp}. \end{align}

Taking the difference between the time average of (4.7) and (4.10), an equation for the perpendicular turbulent kinetic energy is finally obtained

(4.11)\begin{align} & \frac{\partial \bar{n}k_{{\perp}}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\bar{n}k_{{\perp}}\tilde{\boldsymbol{V}}_{C} + \frac{m\overline{n\boldsymbol{V}_0''^2\boldsymbol{V}_{C}''}}{2} + \overline{\boldsymbol{\varPi}\boldsymbol{\cdot}\boldsymbol{V}_{0,\perp}''} + \overline{\phi'\boldsymbol{J}'} + \overline{p'\boldsymbol{V}_{E}'}\right) \nonumber\\ & \quad =\overline{J_{\|}'\nabla_{\|}\phi'} + \overline{p'\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{V}_{E}'} - m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{0,\perp}''}:\boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{0,\perp}^{\rm T} + \overline{\boldsymbol{\varPi}:\boldsymbol{\nabla}\boldsymbol{V}_{0,\perp}''^{\rm T}} - \overline{\boldsymbol{\nabla}_{\perp} p_{{i}}\boldsymbol{\cdot}\boldsymbol{V}_{p}''} \nonumber\\ & \qquad - m\overline{nV_{\|}\frac{{\rm D}\boldsymbol{b}}{{\rm D} t}\boldsymbol{\cdot}\boldsymbol{V}_{0,\perp}''} + \frac{\bar{\boldsymbol{J}}_{p}}{\bar{n}}\boldsymbol{\cdot}\overline{n'\boldsymbol{\nabla}\phi'} - \frac{m\overline{S_{n_{i}}\boldsymbol{V}_{0,\perp}''^2}}{2} - m\tilde{\boldsymbol{V}}_{0,\perp} \boldsymbol{\cdot}\overline{\boldsymbol{V}_{0,\perp}''S_{n_{i}}} + \overline{\boldsymbol{S}_{m}\boldsymbol{\cdot}\boldsymbol{V}_{0,\perp}''}. \end{align}

The manipulations performed in this section and the equations in which they resulted have largely been based on the seminal paper by Scott (Reference Scott2003). However, a clearer definition of the averaging operators has been used here, which consistently includes density fluctuations. Next to changes in the exact form of some terms in terms of averages and fluctuations, this leads to the appearance of the ‘Favre term’ in (4.10) and (4.11) (seventh term on the right-hand side). Furthermore, particle and momentum sources ($S_{n}$ and $\boldsymbol {S}_{m}$) have been retained in the equation set (including their possible fluctuations) as well as the ${\textrm {D}\boldsymbol {b}}/{\textrm {D} t}$ term which were not included in the paper by Scott (Reference Scott2003). Also, we chose to keep the general form of the viscous stress tensor $\boldsymbol{\varPi}$ instead of assuming a particular model for it. On the other hand, magnetic field fluctuations have been neglected here, while they were (partly) maintained in the original paper.

We leave a discussion of the perpendicular kinetic energy equations (4.7), (4.10) and (4.11) and the physics they included for § 4.4.

4.2. Parallel kinetic energy equations

There is also kinetic energy in the parallel ion plasma velocity, both in the mean-field and the fluctuating component. In analogy with the perpendicular kinetic energies, we write the parallel kinetic energies as

(4.12a–c)\begin{equation} E_{{k},\|} \triangleq \frac{m\boldsymbol{V}_{\|}^2}{2},\quad E_{{k},m,\|} \triangleq \frac{m\tilde{\boldsymbol{V}}_{\|}^2}{2},\quad \bar{n} k_{\|} \triangleq \frac{m\overline{n \boldsymbol{V}_{\|}''^2}}{2}. \end{equation}

Transport equations for the parallel kinetic energy can be obtained in a rather straightforward way which is very similar to the typical procedure used in hydrodynamic turbulence. An equation for $E_{{k},\|}$ can be obtained by multiplying the parallel momentum equation (2.23) with $V_{\|}$

(4.13)\begin{align} & \frac{\partial }{\partial t}(nE_{{k},\|}) + \boldsymbol{\nabla}\boldsymbol{\cdot} (n\boldsymbol{V}_{C}E_{{k},\|} + \boldsymbol{\varPi}\boldsymbol{\cdot}\boldsymbol{V}_{\|}) \nonumber\\ & \quad =\boldsymbol{\varPi}:\boldsymbol{\nabla}\boldsymbol{V}_{\|}^{\rm T}- V_{\|}\nabla_{\|} p + mV_{\|} \frac{{\rm D}\boldsymbol{b}}{{\rm D} t}\boldsymbol{\cdot} n\boldsymbol{V}_{0,\perp} - E_{k,\|}S_{n_{i}} + V_{\|}S_{m,\|}. \end{align}

In this derivation, the ion continuity equation (2.1) has been used, in which the convection velocity is assumed to be $\boldsymbol {V}_{C}$, as in the ion momentum equation. Likewise, a mean-field parallel kinetic energy equation is constructed by multiplying the average of (2.23) with $\tilde {V}_{\|}$. This yields

(4.14)\begin{align} & \frac{\partial }{\partial t}(\bar{n}E_{{k},m,\|}) + \boldsymbol{\nabla}\boldsymbol{\cdot} (m\bar{n}\tilde{\boldsymbol{V}}_{C} E_{{k},m,\|} + m\overline{n \boldsymbol{V}_{C}''\boldsymbol{V}_{\|}''}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{\|} + \bar{\boldsymbol{\varPi}}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{\|}) \nonumber\\ & \quad = m\overline{n \boldsymbol{V}_{C}''\boldsymbol{V}_{\|}''}:\boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{\|}^{\rm T} + \bar{\boldsymbol{\varPi}}:\boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{\|}^{\rm T} - \tilde{V}_{\|}\nabla_{\|}\bar{p} + m\tilde{V}_{\|}\overline{\frac{{\rm D}\boldsymbol{b}}{{\rm D} t}\boldsymbol{\cdot} n\boldsymbol{V}_{0,\perp}} \nonumber\\ & \qquad - E_{{k},m,\|}\bar{S}_{n_{i}} + \tilde{V}_{\|}\bar{S}_{m,\|}. \end{align}

Taking the difference between the average of (4.13) and (4.14), an equation for the parallel turbulent kinetic energy is found as

(4.15)\begin{align} & \frac{\partial }{\partial t}(\bar{n}k_{\|}) + \boldsymbol{\nabla}\boldsymbol{\cdot} \left(\bar{n}\tilde{\boldsymbol{V}}_{C} k_{\|} + \frac{m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{\|}''^2}}{2} + \overline{\boldsymbol{\varPi}\boldsymbol{\cdot}\boldsymbol{V}_{\|}''}\right) \nonumber\\ & \quad =- m\overline{n \boldsymbol{V}_{C}''\boldsymbol{V}_{\|}''}:\boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{\|}^{\rm T} + \overline{\boldsymbol{\varPi}:\boldsymbol{\nabla} \boldsymbol{V}_{\|}''^{\rm T}} - \overline{V_{\|}''\nabla_{\|} p} + m\overline{V_{\|}''\frac{{\rm D}\boldsymbol{b}}{{\rm D} t}\boldsymbol{\cdot} n\boldsymbol{V}_{0,\perp}} \nonumber\\ & \qquad - \frac{\overline{S_{n_{i}}V_{\|}''^2}}{2} - \tilde{V}_{\|}\overline{V_{\|}'' S_{n_{i}}} + \overline{V_{\|}''S_{m,\|}}. \end{align}

Beside the $\textrm {D}\boldsymbol {b}/\textrm {D} t$ terms, the interpretation of these equations is rather standard and deviates little from that for the kinetic energy in hydrodynamic turbulence. We will come back to this interpretation in § 4.4.

4.3. Magnetic energy equations

As mentioned above, the magnetic field can be described in terms of the magnetic vector potential

(4.16)\begin{equation} \boldsymbol{B} = \boldsymbol{\nabla} \times\boldsymbol{A} = \boldsymbol{\nabla} \times\boldsymbol{A}_{\|} + \boldsymbol{\nabla} \times\boldsymbol{A}_{{\perp}}. \end{equation}

The magnetic energy can then be decomposed as

(4.17)\begin{equation} \frac{\boldsymbol{B}^2}{2\mu} = \frac{(\boldsymbol{\nabla} \times\boldsymbol{A}_{\|})^2}{2\mu} + \frac{(\boldsymbol{\nabla} \times\boldsymbol{A}_{{\perp}})^2}{2\mu} + \frac{\nabla_{\|}\boldsymbol{A}_{{\perp}}\boldsymbol{\cdot}\boldsymbol{\nabla}_{\perp}A_{\|}}{\mu}, \end{equation}

where the last term enters through reworking the scalar product between $\boldsymbol {\nabla } \times \boldsymbol {A}_{\|}$ and $\boldsymbol {\nabla } \times \boldsymbol {A}_{\perp }$. In this work, we will only study the first term. The dynamics of the second contribution is not included in the equation set considered here, i.e. $\boldsymbol {A}_{\perp }$ is in equilibrium at the time scales of interest. The third term could be assumed to be small on the hypothesis that the parallel gradient of the equilibrium quantity $\boldsymbol {A}_{\perp }$ is small.

Hence, we define the (relevant part of the) total, mean-field and turbulent magnetic energies as

(4.18a–c)\begin{equation} E_{B} \triangleq \frac{(\boldsymbol{\nabla} \times\boldsymbol{A}_{\|})^2}{2\mu},\quad E_{B,m} \triangleq \frac{(\boldsymbol{\nabla} \times\bar{\boldsymbol{A}}_{\|})^2}{2\mu},\quad k_{B} \triangleq \frac{\overline{(\boldsymbol{\nabla} \times\boldsymbol{A}_{\|}')^2}}{2\mu}. \end{equation}

Note that a regular Reynolds decomposition is used in these definitions since the magnetic energy per unit volume is independent of the density. Remark also that the sum of mean-field and turbulent magnetic energy equals the averaged total magnetic energy

(4.19)\begin{equation} \overline{E_{B}}=E_{B,m} + k_{B}. \end{equation}

Following Scott (Reference Scott2003), an equation for the total magnetic energy is derived by taking the product of the parallel current with Ohm's law (2.21) divided by $en$

(4.20)\begin{equation} J_{\|}\frac{\partial A_{\|}}{\partial t} = J_{\|}\frac{\nabla_{\|}p_{{e}}}{en} - J_{\|}\nabla_{\|}\phi - J_{\|}\frac{R_{ei,\|}}{en}. \end{equation}

The left-hand side can be rewritten using (2.22) to find

(4.21)\begin{equation} J_{\|}\frac{\partial A_{\|}}{\partial t} = \frac{1}{2\mu} \frac{\partial}{\partial t}(\boldsymbol{\nabla}_{\perp}A_{\|})^2 - \frac{1}{\mu} \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\frac{\partial A_{\|}}{\partial t}\boldsymbol{\nabla}_{\perp}A_{\|}\right)\!. \end{equation}

For the first term in the above equation, it is easy to show that

(4.22)\begin{equation} \frac{(\boldsymbol{\nabla}_{\perp}A_{\|})^2}{2\mu} = \frac{(\boldsymbol{\nabla} \times\boldsymbol{A}_{\|})^2}{2\mu} = E_{B}. \end{equation}

So finally, we get the total magnetic energy equation

(4.23)\begin{equation} \frac{\partial E_{B}}{\partial t} - \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\frac{\partial A_{\|}}{\partial t} \frac{\boldsymbol{\nabla}_{\perp}A_{\|}}{\mu}\right) = J_{\|}\frac{\nabla_{\|}p_{{e}}}{en} - J_{\|}\nabla_{\|}\phi - J_{\|}\frac{R_{ei,\|}}{en}. \end{equation}

Following an exactly analogous derivation using the average parallel current $\bar {J}_{\|}$ and the average Ohm's law (3.10), the mean-field magnetic energy is found as

(4.24)\begin{equation} \frac{\partial E_{B,m}}{\partial t} - \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\frac{\partial\bar{A}_{\|}}{\partial t} \frac{\boldsymbol{\nabla}_{{\perp}}\bar{A}_{\|}}{\mu}\right) = \bar{J}_{\|}\overline{\left(\frac{\nabla_{\|}p_{{e}}}{en} \right)} - \bar{J}_{\|}\nabla_{\|}\bar{\phi} - \bar{J}_{\|}\overline{\left( \frac{R_{ei,\|}}{en} \right)}. \end{equation}

We remark here that $\bar {J}_{\|}\partial \bar {A}_{\|}/\partial t$ is in general non-zero during stationary tokamak operation. This is in fact the power inserted into the plasma due to the electromagnetic induction of the main transformer. This term is decomposed in $\partial E _{B,m}/\partial t$, which would ideally be zero in a stationary state (constant average magnetic field) and the transport term $-\boldsymbol {\nabla }\boldsymbol {\cdot }(\boldsymbol {\nabla }_{\perp }\bar {A}_{\|}/\mu \partial \bar {A}_{\|}/\partial t)$. In § 3, it has already been discussed that $\partial \bar {A}_{\|}/\partial t$ is non-zero in stationary operation. Secondly, $-\boldsymbol {\nabla }_{\perp }\bar {A}_{\|}/\mu$ is related to the total plasma current enclosed by the flux surface on which it is evaluated and should thus be non-zero as well. Hence, $-\boldsymbol {\nabla }_{\perp }\bar {A}_{\|}/\mu \partial \bar {A}_{\|}/\partial t$ can be interpreted as an influx of magnetic energy from the boundary of the computational domain, as induced by the central solenoid.

Taking the difference between the average of (4.23) and (4.24), an equation for the energy in the magnetic field fluctuations is obtained

(4.25)\begin{equation} \frac{\partial k_{B}}{\partial t} - \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\overline{\frac{\partial A_{\|}'}{\partial t} \frac{\boldsymbol{\nabla}_{\perp}A_{\|}'}{\mu}}\right) = \overline{J_{\|}' \left(\frac{\nabla_{\|}p_{{e}}}{en}\right)'} - \overline{J_{\|}'\nabla_{\|}\phi'} - \overline{J_{\|}' \left(\frac{R_{ei,\|}}{en}\right)'}. \end{equation}

4.4. Energy theorem

This section illustrates the energetic couplings between the perpendicular kinetic energy (mean-field (4.10) and turbulent (4.11)), thermal energy (electron (3.9) and ions (3.8)), parallel kinetic energy (mean-field (4.14) and turbulent (4.15)) and magnetic energy (mean-field (4.24) and turbulent (4.25)) and thus of the different pathways for the energy to get into and out of the turbulence. This discussion is largely based on the insight provided by Scott (Reference Scott2003).

Our main interest is the perpendicular turbulent kinetic energy which is supposedly driving the turbulent transport of heat and particles. As such, we first consider the left-hand side of (4.11). The first term is the time rate of change of the turbulent kinetic energy. Then, all terms under the divergence operator represent fluxes transporting turbulent kinetic energy from one location to another. The first flux is the convection of $k_{\perp }$ by the mean-field velocity, the second one the convection by turbulent fluctuations and the third one a flux due to viscous stresses. These terms also appear in regular hydrodynamic turbulence, see for example Pope (Reference Pope2015) and Canuto (Reference Canuto1997). Next, a flux due to electrostatic potential and current fluctuations follows and then a flux of pressure due to $E\times B$ velocity fluctuations. Scott (Reference Scott2003) argues that that $\boldsymbol {\nabla }\boldsymbol {\cdot }(\overline {\phi '\boldsymbol {J}'}+\overline {p'\boldsymbol {V}_{E}'})\approx \boldsymbol {\nabla }\boldsymbol {\cdot }\overline {\phi '\boldsymbol {J}_{\|}'}$, since $p\boldsymbol {V}_{E}$ cancels with $\phi \boldsymbol {J}_{\perp }$ due to Pointing cancellation. Furthermore, it can be shown that $-\boldsymbol {\nabla }\boldsymbol {\cdot }(\phi \boldsymbol {J}_{*} + p\boldsymbol {V}_{E}) = \boldsymbol {\nabla }\boldsymbol {\cdot }( \boldsymbol {\nabla }(\bar {p\phi })\times \boldsymbol {b}/B )$, which vanishes in 1-D radial geometries (Coosemans et al. Reference Coosemans, Dekeyser and Baelmans2021b). The remaining transport due to $\overline {\phi '\boldsymbol {J}_{\|}'}$ thus constitutes a non-convective parallel transport term that does not have an equivalent in hydrodynamic turbulence. Note that all these transport terms, except for mean-field convection, constitute closure terms, and that little quantitative information is available for them a priori. Singh & Diamond (Reference Singh and Diamond2020) focus on the radial transport of turbulence intensity and investigate when the associated terms are important. As far as the authors are aware, the $\overline {\phi '\boldsymbol {J}_{\|}'}$ flux has received very little attention in the literature. However, § 6 will argue it may play an important role in plasma edge turbulence.

Next, the coupling between mean-field and turbulent kinetic energy is considered. As in hydrodynamic turbulence, the Reynolds stress (RS) term (third term on the right-hand side of (4.10) and (4.11), different sign in both) exchanges energy between the turbulent and mean-field kinetic energy. While in 3-D hydrodynamic turbulence this term typically drives the turbulence, in plasma edge physics it is expected to act like a sink of $k_{\perp }$ and thus a source of $E_{k,m,\perp }$ because of the inverse energy cascade. Hence, this term is expected to be important close to the separatrix, where strong shear flows tend to develop, which are partly fed by the turbulent kinetic energy by tearing apart small eddies. Moreover, this term is also expected to play an important role in the generation of these shear flows (Diamond et al. Reference Diamond, Itoh, Itoh and Hahm2005; Manz et al. Reference Manz, Xu, Fedorczak, Thakur and Tynan2012; Held et al. Reference Held, Wiesenberger, Kube and Kendl2018). However, when the turbulence is sufficiently damped and when the flow shear is sufficiently strong, the RS term may act like a source of $k_{\perp }$ through the Kelvin–Helmholtz (KH) instability (Rogers & Dorland Reference Rogers and Dorland2005; Myra et al. Reference Myra, D'Ippolito, Russell, Umansky and Baver2016; Giacomin & Ricci Reference Giacomin and Ricci2020). In addition, the ‘Favre term’, seventh term on the right-hand side in both equations, also exchanges energy between mean-field and turbulent kinetic energy. This term appeared due to the non-commutativity of the gradient and Favre averaging operators. Thus, this term only appeared in the equation because of the rigorous average techniques that have been applied and had not been identified in the literature before as far as the authors are aware. The RSs clearly constitutes a closure term. The Favre term contains closure terms in $\bar {\boldsymbol {J}}_{p}$, while the $\overline {n'\boldsymbol {\nabla }\phi '}$ factor can be calculated from $\boldsymbol {\varGamma }_{n,t,E}$.

Comparing the right-hand side of the thermal energy equations (3.8) and (3.9) with that of the kinetic energy equations (4.10) and (4.11), it can be seen that interchange terms involving $p\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {V}_{E}$ appear in all of them. Hence, the fluctuations of this interchange term exchange energy between the turbulence and the thermal energy. This energy transfer could occur in both directions (from thermal energy to turbulence and vice versa), see § 6. However, it is expected that this interchange term will provide an important source of the turbulence on the outboard side of the tokamak, especially in the SOL (Scott Reference Scott2003, Reference Scott2005; Ribeiro & Scott Reference Ribeiro and Scott2005). While the interchange term is a closure term, an analytical model for it has been found; see these same references and § 6. The viscous stress and $\nabla_{\perp}p _{{i}}\boldsymbol {\cdot }\boldsymbol {V}_{p}$ terms likewise exchange energy between the kinetic energy (both mean-field and turbulent) and ion thermal energy. The former acts like a sink dissipating kinetic energy into thermal energy. Coosemans et al. (Reference Coosemans, Dekeyser and Baelmans2021b, Reference Coosemans, Dekeyser and Baelmans2022) found it to be of secondary importance for the $k_{\perp }$ balance for simulations with the TOKAM2D turbulence code (Sarazin & Ghendrih Reference Sarazin and Ghendrih1998; Moulton et al. Reference Moulton, Marandet, Tamain, Ghendrih and Futtersack2014; Baudoin et al. Reference Baudoin, Tamain, Ciraolo, Futtersack, Gallo, Ghendrih, Marandet, Nace and Norscini2016; Marandet et al. Reference Marandet, Nace, Valentinuzzi, Tamain, Bufferand, Ciraolo, Genesio and Mellet2016; Nace Reference Nace2018). Further research is required to quantify the importance of the latter term. Again, both terms appear as closure terms in the $k_{\perp }$ equation.

Then, the terms involving $J_{\|}\nabla_{\|}\phi$ in the $E_{k,m,\perp }$ and $k_{\perp }$ equation can be shown to exchange energy with the mean-field and turbulent magnetic energy ((4.24) and (4.25) respectively). Furthermore, the other two terms on the right-hand side of (4.24) and (4.25) exchange energy between this magnetic energy and the electron thermal energy in (3.9). The last term in the magnetic energy equations is due to electron–ion friction and is expected to dissipate magnetic energy, providing a unidirectional transfer to the thermal energy. The first term on the right-hand side of the equation is assumed to allow energy transfer from thermal energy to magnetic energy and from there to the turbulent kinetic energy. Hence, this constitutes a second channel by which energy can be injected from the thermal energy into the turbulence (next to the interchange channel) (Scott Reference Scott2003, Reference Scott2005). This transfer channel is related to the dynamics parallel to the magnetic field and especially the drift-wave (DW) instability is expected to act on this channel. Because the parallel dynamics has more freedom to evolve in the closed field line region than in the SOL, it is expected to be especially important there (Scott Reference Scott2003, Reference Scott2005; Ribeiro & Scott Reference Ribeiro and Scott2005).

Note that, in the electrostatic case in which $A_{\|}'=0$, the left-hand side of (4.25) vanishes. Nonetheless, the DW coupling remains active, be it in a slightly simplified form. The terms $-\overline {J_{\|}'(\nabla_{\|}p_{{e}}/(ne))'}+\overline {J_{\|}'(R_{ei,\|}/(n e))'}$ in the electron thermal energy equation (3.9) could then be replaced by $-\overline {J_{\|}'\nabla_{\|}\phi '}$, such that the coupling is directly between the thermal energy and the turbulent kinetic energy (without the mediation of the magnetic energy). Furthermore we remark that the left-hand side of the mean-field magnetic energy equation (4.24) actually represents the power induced in the plasma through electromagnetic induction by the central solenoid. As this power is injected into the magnetic energy, it could in theory not just be dissipated through the resistive term, but may also find its way to the mean-field kinetic energy through the $\bar {J}_{\|}\nabla_{\|}\bar {\phi }$ term.

Next, the $\textrm {D}\boldsymbol {b}/\textrm {D} t$ terms in the perpendicular kinetic energy equations (4.10) and (4.11) exchange energy with the parallel kinetic energies in (4.14) and (4.15). These terms are due to changes in the magnetic field direction. Notice that the $\textrm {D}\boldsymbol {b}/\textrm {D} t$ term in the $k_{\perp }$ equation not only exchanges energy with $k_{\|}$, but also with $E_{{k},m,\|}$. Likewise, this term allows energy transfer from $E_{k,m,\perp }$ to $E_{{k},m,\|}$ and $k_{\|}$. Note that these terms only appear in the perpendicular kinetic energy equations because the $\boldsymbol {J}_{p,\|}$ contribution to the polarisation current (defined in (2.14)) has been retained (which is why they were not included in Scott Reference Scott2003). The $\textrm {D}\boldsymbol {b}/\textrm {D} t$ contributions to the $\boldsymbol {J}_{p,E}$ and $\boldsymbol {J}_{p,*}$ (see definitions (2.15) and (2.16)) to the contrary do not individually lead to additional terms in any of the energy equations. The importance of these energy transfer channels is unknown. It may be expected that their magnitude and sign depend on the direction of the magnetic field in the reactor (forward field or reverse field). Note that these terms represent the only direct coupling between parallel and perpendicular kinetic energies. However, energy could also be transferred between them by the mediation of the ion and electron thermal energy. Furthermore, it is very possible that the parallel kinetic energy dynamics implicitly impacts the turbulence, for example by its impact on or competition with other energy transfer channels.

The interpretation of the other terms in the parallel kinetic energy equations (4.14) and (4.15) is rather standard and deviates little from that for the kinetic energy in hydrodynamic turbulence. The terms on the left-hand side under the divergence represent kinetic energy fluxes due to mean-field convection, turbulent convection and viscous transport. On the right-hand side, the first term is the viscous dissipation of kinetic energy which is converted into ion thermal energy. The second term contains RSs acting on the gradients of the parallel (ion) velocity and exchanges energy between turbulent and mean-field parallel kinetic energy again. The third term is the work done by the pressure gradient on the parallel velocity. Both for mean-field and turbulent energy this exchanges energy with the ion and electron thermal energy. This energy transfer might again take place in either direction. The last three terms are due to particle and momentum sources, which are expected to be mainly due to plasma–neutral interactions.

Finally, the last three terms on the right-hand side of the perpendicular turbulent kinetic energy equation (4.11) are due to particle and momentum sources in the plasma. The prime particle and momentum sources in the plasma are expected to be due to the ionisation of neutral particles and collisions with neutrals. These source terms are again closure terms, about which very limited information is available a priori.

Now, all source/sink terms on the right-hand side of the energy equations (3.8), (3.9), (4.10), (4.11), (4.14), (4.15), (4.24) and (4.25) appear in two equations with opposite signs, hence conserving their total energy. The energetic couplings between the different energy forms in plasma edge turbulence are illustrated in figure 1. The figure aims to give an overview of some of the couplings which are expected to be important, but does not contain all the energy transfer channels. This figure is inspired by a similar figure in Scott (Reference Scott2003).

Figure 1.

Schematic representation of the main energy transfer channels between the different energy forms in plasma edge turbulence. Adapted from Scott (Reference Scott2003).

5.1. The $E\times B$-only kinetic energy equations

In analogy to the total kinetic energies, we define the $E\times B$-only kinetic energies as

(5.1a–c)\begin{equation} E_{{E}} \triangleq \frac{m\boldsymbol{V}_{E}^2}{2},\quad E_{E,m} \triangleq \frac{m\tilde{\boldsymbol{V}}_{E}^2}{2},\quad \bar{n} k_{E} \triangleq \frac{m\overline{n \boldsymbol{V}_{E}''^2}}{2}. \end{equation}

An equation for $E_{{E}}$ is likewise derived starting from the charge balance equation (4.3), as was done to obtain the equation for $E_{{k},\perp }$. However, the diamagnetic drift contribution to the polarisation current is now kept separate in the equivalent to (4.4)

(5.2)\begin{align} \phi\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{J}_{p} = \boldsymbol{\nabla}\boldsymbol{\cdot}\phi\boldsymbol{J}_{p} - \boldsymbol{\nabla}\phi\boldsymbol{\cdot}\boldsymbol{J}_{p} = \boldsymbol{\nabla}\boldsymbol{\cdot}\phi\boldsymbol{J}_{p} - \boldsymbol{\nabla}\phi\boldsymbol{\cdot}(\boldsymbol{J}_{p,E} + \boldsymbol{J}_{p,\|} + \boldsymbol{J}_{p,\varPi}) - \boldsymbol{\nabla}\phi\boldsymbol{\cdot}\boldsymbol{J}_{p,*}, \end{align}

where the decomposition of $\boldsymbol {J}_{p}$ introduced in (2.9) and (2.13) is used. In analogy to expression (4.5), the second term on the right-hand side in the previous expression yields the time rate of change of the $E\times B$-only kinetic energy

(5.3)\begin{align} & -\boldsymbol{\nabla}_{{\perp}}\phi\boldsymbol{\cdot}(\boldsymbol{J}_{p,E} + \boldsymbol{J}_{p,\varPi} + \boldsymbol{J}_{p,\|}) = \frac{\partial nE_{{E}}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}(nE_{{E}}\boldsymbol{V}_{C} + \boldsymbol{\varPi}\boldsymbol{\cdot}\boldsymbol{V}_{E}) \nonumber\\ & \quad - \boldsymbol{\varPi}:\boldsymbol{\nabla}\boldsymbol{V}_{E}^{\rm T} + mnV_{\|}\frac{{\rm D}\boldsymbol{b}}{{\rm D} t}\boldsymbol{\cdot}\boldsymbol{V}_{E} + E_{E}S_{n_{i}} - \boldsymbol{S}_{m}\boldsymbol{\cdot}\boldsymbol{V}_{E}. \end{align}

Filling out expressions (5.2)–(5.3) together with the manipulation (4.6) derived before into (4.3), the equation for $E_{{E}}$ is finally obtained

(5.4)\begin{align} & \frac{\partial nE_{{E}}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot} (nE_{{E}}\boldsymbol{V}_{C} + \boldsymbol{\varPi}\boldsymbol{\cdot}\boldsymbol{V}_{E} + \phi\boldsymbol{J} + p\boldsymbol{V}_{E}) = J_{\|}\nabla_{\|}\phi +p\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{V}_{E} + \boldsymbol{\varPi}:\boldsymbol{\nabla}\boldsymbol{V}_{E}^{\rm T} \nonumber\\ & \quad + \boldsymbol{\nabla}\phi\boldsymbol{\cdot}\boldsymbol{J}_{p,*} - mnV_{\|}\frac{{\rm D}\boldsymbol{b}}{{\rm D} t} \boldsymbol{\cdot}\boldsymbol{V}_{E} - E_{{E}} S_{n_{i}} + \boldsymbol{S}_{m}\boldsymbol{\cdot}\boldsymbol{V}_{E}. \end{align}

The form of this equation is very similar to (4.7). However, $E_{{k},\perp }$ is ‘replaced’ by $E_{{E}}$ and $\boldsymbol {V}_{0,\perp }$ by $\boldsymbol {V}_{E}$. Furthermore, the $\boldsymbol {\nabla }p_{{i}}\boldsymbol {\cdot }\boldsymbol {V}_{p}$ term is no longer present in the equation, while the last term on the right-hand side of (5.2) comes in to include the diamagnetic drift contribution to the polarisation current. Thus the latter is not neglected. Instead, it is chosen to account for it as a current divergence term instead of including it in the kinetic energy on the left-hand side, as was done in the total kinetic energy case in § 4.1. When the diamagnetic drift contribution to the polarisation current is small or is neglected, the $\phi \boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {J}_{p,*}$ term can also be neglected. On the other hand, it is worth noting that the DW and interchange terms remain unchanged between the total perpendicular and the $E\times B$-only kinetic energy equations. Note that (5.4) could of course also have been derived directly from (4.7) by algebraic manipulation. This derivation proved, however, much more tortuous and tedious.

A mean-field $E\times B$-only kinetic energy equation is derived by starting from (4.8) again. Applying analogous averaging operations as used in § 4.1 to the derivation made just before for the total $E\times B$ kinetic energy, we find

(5.5)\begin{align} & \frac{\partial \bar{n}E_{E,m}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}(\bar{n}E_{E,m}\tilde{\boldsymbol{V}}_{C} + m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{E}''}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{E} + \bar{\boldsymbol{\varPi}}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{E} + \bar{\phi}\bar{\boldsymbol{J}} + \bar{p}\bar{\boldsymbol{V}}_{E}) \nonumber\\ & \quad =\bar{J}_{\|}\nabla_{\|}\bar{\phi} + \bar{p}\boldsymbol{\nabla}\boldsymbol{\cdot}\bar{\boldsymbol{V}}_{E} + m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{E}''}: \boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{E}^{\rm T} + \bar{\boldsymbol{\varPi}}:\boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{E}^{\rm T} + \boldsymbol{\nabla}\bar{\phi}\boldsymbol{\cdot}\bar{\boldsymbol{J}}_{p,*} \nonumber\\ & \qquad -m\overline{nV_{\|}\frac{{\rm D}\boldsymbol{b}}{{\rm D} t}}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{E} -\left(\frac{\bar{\boldsymbol{J}}_{p,E}+\bar{\boldsymbol{J}}_{p,\varPi}}{\bar{n}}\right) \boldsymbol{\cdot}\overline{n'\boldsymbol{\nabla}\phi'} - E_{E,m}\bar{S}_{n_{i}} + \bar{\boldsymbol{S}}_{m}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{E}. \end{align}

Taking the difference between the average of (5.4) and (5.5) again yields an equation for the turbulent kinetic energy, in the $E\times B$ drift fluctuations this time

(5.6)\begin{align} & \frac{\partial \bar{n}k_{E}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\bar{n}k_{E}\tilde{\boldsymbol{V}}_{C} + \frac{m\overline{n\boldsymbol{V}_{E}''^2\boldsymbol{V}_{C}''}}{2} + \overline{\boldsymbol{\varPi}\boldsymbol{\cdot}\boldsymbol{V}_{E}''} + \overline{\phi'\boldsymbol{J}'} + \overline{p'\boldsymbol{V}_{E}'}\right)= \overline{J_{\|}'\nabla_{\|}\phi'} \nonumber\\ & \quad + \overline{p'\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{V}_{E}'} - m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{E}''}:\boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{E}^{\rm T} + \overline{\boldsymbol{\varPi}:\boldsymbol{\nabla}\boldsymbol{V}_{E}''^{\rm T}} + \overline{\boldsymbol{\nabla}\phi'\boldsymbol{\cdot}\boldsymbol{J}_{p,*}'} - m\overline{nV_{\|}\frac{{\rm D}\boldsymbol{b}}{{\rm D} t}\boldsymbol{\cdot}\boldsymbol{V}_{E}''} \nonumber\\ & \quad + \left(\frac{\bar{\boldsymbol{J}}_{p,E}+\bar{\boldsymbol{J}}_{p,\varPi}}{\bar{n}}\right)\boldsymbol{\cdot}\overline{n'\boldsymbol{\nabla}\phi'} - \frac{m\overline{S_{n_{i}}\boldsymbol{V}_{E}''^2}}{2} - m\tilde{\boldsymbol{V}}_{E}\boldsymbol{\cdot}\overline{\boldsymbol{V}_{E}''S_{n_{i}}} + \overline{\boldsymbol{S}_{m}\boldsymbol{\cdot}\boldsymbol{V}_{E}''}. \end{align}

Comparing these $E\times B$-only kinetic energy equations (5.5) and (5.6) with the total perpendicular kinetic energy equations (4.10) and (4.11), the same changes as for the total kinetic energy can be observed. In addition, it is worth remarking that the RSs (second term on the right-hand side) have also changed now, in the sense that these now include only the $E\times B$ velocity (and $\boldsymbol {V}_{C}$). Likewise, in the Favre averaging term (seventh term on the right-hand side) the diamagnetic drift polarisation current is no longer present.

5.2. Ion-diamagnetic-drift-only kinetic energy equations

To complement the equations for the $E\times B$-energy and to deepen the energy theorem, we derive equations for the kinetic energy in the ion diamagnetic drift in this section and in the ‘mixed’ $E\times B$-ion diamagnetic kinetic energy in the next section. The total, mean-field and turbulent diamagnetic kinetic energies are defined as

(5.7a–c)\begin{equation} E_{*} \triangleq \frac{m\boldsymbol{V}_{*,i}^2}{2},\quad E_{*,m} \triangleq \frac{m\tilde{\boldsymbol{V}}_{*,i}^2}{2},\quad \bar{n}k_{*} \triangleq \frac{m\overline{n \boldsymbol{V}_{*,i}''^2}}{2}. \end{equation}

In analogy to (4.5) and (5.3), a transport relation for the total kinetic energy in the diamagnetic drift velocity can be obtained as

(5.8)\begin{align} & -\frac{\boldsymbol{\nabla}_{\perp} p_{{i}}}{en}\boldsymbol{\cdot}(\boldsymbol{J}_{p,*} + \boldsymbol{J}_{p,\varPi} + \boldsymbol{J}_{p,\|}) = \frac{\partial nE_{*}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}( nE_{*}\boldsymbol{V}_{C} + \boldsymbol{\varPi}\boldsymbol{\cdot}\boldsymbol{V}_{*,i} ) \nonumber\\ & \quad -\boldsymbol{\varPi}:\boldsymbol{\nabla}\boldsymbol{V}_{*,i}^{\rm T} + mnV_{\|}\frac{{\rm D}\boldsymbol{b}}{{\rm D} t}\boldsymbol{\cdot}\boldsymbol{V}_{*,i} + E_{*} S_{n,i} - \boldsymbol{S}_{m}\boldsymbol{\cdot}\boldsymbol{V}_{*,i}. \end{align}

Slightly rewriting readily yields the total diamagnetic kinetic energy equation as

(5.9)\begin{align} & \frac{\partial nE_{*}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}( nE_{*}\boldsymbol{V}_{C} + \boldsymbol{\varPi}\boldsymbol{\cdot}\boldsymbol{V}_{*,i} ) = \boldsymbol{\varPi}:\boldsymbol{\nabla}\boldsymbol{V}_{*,i}^{\rm T} - \boldsymbol{\nabla}_{\perp} p_{{i}}\boldsymbol{\cdot}(\boldsymbol{V}_{p,*}+\boldsymbol{V}_{p,\varPi} + \boldsymbol{V}_{p,\|}) \nonumber\\ & \quad - mnV_{\|}\frac{{\rm D}\boldsymbol{b}}{{\rm D} t}\boldsymbol{\cdot}\boldsymbol{V}_{*,i} - E_{*} S_{n,i} + \boldsymbol{S}_{m}\boldsymbol{\cdot}\boldsymbol{V}_{*,i}. \end{align}

An analogous derivation starting from $\boldsymbol {\nabla }_{\perp }\bar {p}_i\boldsymbol {\cdot }(\tilde {\boldsymbol {V}}_{p,*}+\tilde {\boldsymbol {V}}_{p,\varPi } + \tilde {\boldsymbol {V}}_{p,\|})$ yields an equation for the mean-field kinetic energy in the diamagnetic drift velocity

(5.10)\begin{align} & \frac{\partial \bar{n}E_{*,m}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}(\bar{n}E_{*,m}\tilde{\boldsymbol{V}}_{C} + m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{*,i}''}\boldsymbol{\cdot} \tilde{\boldsymbol{V}}_{*,i} + \bar{\boldsymbol{\varPi}}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{*,i}) \nonumber\\ & \quad = m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{*,i}''}:\boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{*,i}^{\rm T} + \bar{\boldsymbol{\varPi}}: \boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{*,i}^{\rm T} -\boldsymbol{\nabla}_{{\perp}}\bar{p}_i\boldsymbol{\cdot}(\tilde{\boldsymbol{V}}_{p,*} + \tilde{\boldsymbol{V}}_{p,\varPi} + \tilde{\boldsymbol{V}}_{p,\|}) \nonumber\\ & \qquad - m\overline{nV_{\|}\frac{{\rm D}\boldsymbol{b}}{{\rm D} t}}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{*,i} - E_{*,m} \bar{S}_{n_{i}} + \bar{\boldsymbol{S}}_{m}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{*,i} . \end{align}

Taking the difference between the average of (5.9) and (5.10), an equation for the turbulent diamagnetic kinetic energy is obtained

(5.11)\begin{align} & \frac{\partial\bar{n}k_{*}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\bar{n}k_{*}\tilde{\boldsymbol{V}}_{C} + \frac{m\overline{n\boldsymbol{V}_{*,i}''^2\boldsymbol{V}_{C}''}}{2} + \overline{\boldsymbol{\varPi}\boldsymbol{\cdot}\boldsymbol{V}_{*,i}''}\right) \nonumber\\ & \quad =- m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{*,i}''}:\boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{*,i}^{\rm T} + \overline{\boldsymbol{\varPi}:\boldsymbol{\nabla}\boldsymbol{V}_{*,i}''^{\rm T}} - \overline{\boldsymbol{\nabla}_{\perp} p_{{i}}\boldsymbol{\cdot}(\boldsymbol{V}_{p,*}''+\boldsymbol{V}_{p,\varPi}'' + \boldsymbol{V}_{p,\|}'')} \nonumber\\ & \qquad - m\overline{nV_{\|}\frac{{\rm D}\boldsymbol{b}}{{\rm D} t}\boldsymbol{\cdot}\boldsymbol{V}_{*,i}''} - \frac{m\overline{S_{n_{i}}\boldsymbol{V}_{*,i}''^2}}{2} - m\tilde{\boldsymbol{V}}_{*,i}\boldsymbol{\cdot}\overline{\boldsymbol{V}_{*,i}''S_{n_{i}}} + \overline{\boldsymbol{S}_{m}\boldsymbol{\cdot}\boldsymbol{V}_{*,i}''}. \end{align}

5.3. Mixed kinetic energy equations

At this point, it is important to remark that the sum of the $E\times B$-only kinetic energy (see definition (5.1a–c)) and the ion-diamagnetic-drift-only kinetic energy (see definition (5.7a–c)) is not equal to the total perpendicular kinetic energy (see definition (4.1a–c)). Decomposing the total perpendicular kinetic energy as

(5.12)\begin{align} E_{{k},\perp} \triangleq \frac{m\boldsymbol{V}_{0,\perp}^2}{2} & = \frac{m(\boldsymbol{V}_{E}+\boldsymbol{V}_{*,i})^2}{2} \nonumber\\ & = \frac{m\boldsymbol{V}_{E}^2}{2} + \frac{m\boldsymbol{V}_{*,i}^2}{2} + m\boldsymbol{V}_{E}\boldsymbol{\cdot}\boldsymbol{V}_{*,i} \triangleq E_{{E}} + E_{*} + E_{\mathrm{mix}}, \end{align}

it is noted that there is a kinetic energy contribution from the ‘mixed’ $E\times B$-ion diamagnetic kinetic energy. We define this mixed kinetic energy as

(5.13a–c)\begin{equation} E_{\mathrm{mix}} \triangleq m\boldsymbol{V}_{E}\boldsymbol{\cdot}\boldsymbol{V}_{*,i},\quad E_{\mathrm{mix},m} \triangleq m\tilde{\boldsymbol{V}}_{E}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{*,i},\quad \bar{n} k_{\mathrm{mix}} \triangleq m\overline{n\boldsymbol{V}_{E}''\boldsymbol{\cdot}\boldsymbol{V}_{*,i}''}. \end{equation}

Note that these mixed kinetic energies can be negative when $\boldsymbol {V}_{E}$ and $\boldsymbol {V}_{*,i}$ are counter-aligned. As such, the term ‘energy’ might strictly speaking be somewhat of a misnomer.

According to (5.12), an equation for the total mixed kinetic energy can be calculated as the difference between (4.7) and (5.4) and (5.9). Likewise, an equation for the mean-field mixed kinetic energy is obtained from the difference between (4.10) and (5.5) and (5.10), and for the turbulent mixed kinetic energy from the difference between equation (4.11) and equations (5.6) and (5.11). This then yields

(5.14)\begin{align} & \frac{\partial nE_{\mathrm{mix}}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot} nE_{\mathrm{mix}}\boldsymbol{V}_{C} =-\boldsymbol{\nabla}_{\perp} p_{{i}}\boldsymbol{\cdot}\boldsymbol{V}_{p,E} - \boldsymbol{\nabla}\phi\boldsymbol{\cdot}\boldsymbol{J}_{p,*} - E_{\mathrm{mix}} S_{n_{i}}, \end{align}

(5.15)\begin{align} & \frac{\partial \bar{n}E_{\mathrm{mix},m}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot} (\bar{n}E_{\mathrm{mix},m}\boldsymbol{V}_{C} + m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{E}''}\boldsymbol{\cdot} \tilde{\boldsymbol{V}}_{*,i}+ m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{*,i}''}\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{E}) \nonumber\\ & \quad = m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{E}''}:\boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{*,i}^{\rm T} + m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{*,i}''}: \boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{E}^{\rm T} -\boldsymbol{\nabla}_{{\perp}}\bar{p}_i\boldsymbol{\cdot}\tilde{\boldsymbol{V}}_{p,E} \nonumber\\ & \qquad - \boldsymbol{\nabla}\bar{\phi}\boldsymbol{\cdot}\bar{\boldsymbol{J}}_{p,*} - e\tilde{\boldsymbol{V}}_{p,*}\boldsymbol{\cdot} \overline{n'\boldsymbol{\nabla}\phi'} - E_{\mathrm{mix},m} \bar{S}_{n_{i}}, \end{align}

(5.16)\begin{align} & \frac{\partial\bar{n} k_{\mathrm{mix}}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}(\bar{n} k_{\mathrm{mix}}\tilde{\boldsymbol{V}}_{C} + m\overline{n\boldsymbol{V}_{E}''\boldsymbol{\cdot}\boldsymbol{V}_{*,i}''\boldsymbol{V}_{C}''}) \nonumber\\ & \quad =- m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{E}''}:\boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{*,i}^{\rm T} - m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{*,i}''}: \boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{E}^{\rm T} - \overline{\boldsymbol{\nabla}_{\perp} p_{{i}}\boldsymbol{\cdot}\boldsymbol{V}_{p,E}''}- \overline{\boldsymbol{\nabla}\phi'\boldsymbol{\cdot}\boldsymbol{J}_{p,*}'} \nonumber\\ & \qquad + e\tilde{\boldsymbol{V}}_{p,*}\boldsymbol{\cdot}\overline{n'\boldsymbol{\nabla}\phi'} - m\overline{S_{n_{i}}\boldsymbol{V}_{E}''\boldsymbol{\cdot}\boldsymbol{V}_{*,i}''} - m\tilde{\boldsymbol{V}}_{E}\boldsymbol{\cdot}\overline{\boldsymbol{V}_{*,i}''S_{n_{i}}} - m\tilde{\boldsymbol{V}}_{*,i}\boldsymbol{\cdot}\overline{\boldsymbol{V}_{E}''S_{n_{i}}}. \end{align}

5.4. Refined energy theorem

The sources and sinks on the right-hand side of (5.4)–(5.6), (5.9)–(5.11) and (5.14)–(5.16) again represent the energetic couplings between the different energy equations. Firstly, this shows that the interchange and DW terms really transfer energy with the $E\times B$ kinetic energy only, and not with the other perpendicular kinetic energy contributions. The viscous stresses, on the other hand, transfer energy between the ion thermal energy (3.8) and the $E\times B$-only and diamagnetic kinetic energy individually, acting on the respective velocities of the latter. The kinetic energy source due to momentum sources and the $\textrm {D}\boldsymbol {b}/\textrm {D} t$ terms are likewise split between both forms of kinetic energy. The kinetic energy source due to the particle source acts independently on all three kinds of kinetic energy.

In addition, the pressure work on the polarisation velocity exchanges energy between the ion thermal energy and $E_{*}$ and $E_{\mathrm {mix}}$ separately. Thus, it can be seen that, somewhat surprisingly, the diamagnetic kinetic energy equation is not directly coupled to the other perpendicular kinetic energy equations at all. Then, the $\boldsymbol {\nabla }\phi \boldsymbol {\cdot }\boldsymbol {J}_{p,*}$ term exchanges energy between $E_{{E}}$ and $E_{\mathrm {mix}}$. Thus, the latter ‘mixed’ form of kinetic energy exchanges energy with the ion thermal energy and the $E\times B$ kinetic energy.

Lastly, the various forms of the RS and Favre terms exchange energy between the mean-field kinetic energy equation and the corresponding turbulent kinetic energy equation.

The fact that the interchange and DW mechanisms, generally assumed to be the main turbulence drives in the plasma edge, exchange energy with the $E\times B$ energy only, seems to be in line with the $E\times B$ fluctuations being stronger than the diamagnetic ones. Moreover, the diamagnetic and mixed kinetic energies have much less (direct) couplings, especially to the thermal energy reservoirs which are supposedly the most important ones. This could be expected to further limit the strength of the turbulence in these components. It is particularly interesting to observe that only the $E\times B$ kinetic energy is coupled to the electron thermal energy and to the magnetic energy.

Figure 2 shows a schematic representation of the energy couplings split out over the different components of the perpendicular kinetic energy. Note that the contributions presumed to be important that were shown in figure 1 are actually all between the $E\times B$-only kinetic energy and other energy forms, never involving the diamagnetic or mixed kinetic energy. As such, we do not put the mathematical expressions for the couplings of the $E\times B$-only kinetic energy (indicated with green arrows) in figure 2 since they were already denoted in figure 1.

Figure 2.

Schematic representation of the energy transfer channels between the different perpendicular kinetic energy forms in plasma edge turbulence.

6.1. Modelling mean-field turbulent fluxes

Section 3 established that the average turbulent $E\times B$ particle flux $\boldsymbol {\varGamma }_{n,t,E}$ in the average continuity equation (3.6) or (3.7) and heat fluxes $\boldsymbol {\varGamma }_{p_{{i}},t,E}$ and $\boldsymbol {\varGamma }_{p_{{e}},t,E}$ in the thermal energy equations (3.8) and (3.9) are vital closure terms to be modelled. Following the common practise in mean-field transport modelling (LaBombard et al. Reference LaBombard, Umansky, Boivin, Goetz, Hughes, Lipschultz, Mossessian, Pitcher and Terry2000; Aho-Mantila et al. Reference Aho-Mantila, Wischmeier, Müller, Potzel, Coster, Bonnin and Conway2012; Reimold et al. Reference Reimold, Wischmeier, Bernert, Potzel, Coster, Bonnin, Reiter, Meisl, Kallenbach and Aho-Mantila2015; Dekeyser et al. Reference Dekeyser, Bonnin, Lisgo, Pitts, Brunner, LaBombard and Terry2016), Coosemans et al. (Reference Coosemans, Dekeyser and Baelmans2020, Reference Coosemans, Dekeyser and Baelmans2021a,Reference Coosemans, Dekeyser and Baelmansb, Reference Coosemans, Dekeyser and Baelmans2022) suggested to model these fluxes through diffusion relations

(6.1a–c)\begin{equation} \boldsymbol{\varGamma}_{n,t,E} \approx-D\boldsymbol{\nabla}_{{\perp}}\bar{n},\quad \boldsymbol{\varGamma}_{p_{{i}},t,E} \approx-\chi_{i}\bar{n}\boldsymbol{\nabla}_{{\perp}}\tilde{T}_i,\quad \boldsymbol{\varGamma}_{p_{{e}},t,E} \approx-\chi_{e}\bar{n}\boldsymbol{\nabla}_{{\perp}}\tilde{T}_e ,\end{equation}

and to relate the effective turbulent transport coefficients $D$, $\chi _i$ and $\chi _e$ to characteristic quantities for the turbulence, i.e. $D=D(k_{E},\zeta _{E},\ldots )$, $\chi _e=\chi _e(k_{E},\zeta _{E},\ldots )$, $\chi _i=\chi _i(k_{E},\zeta _{E},\ldots )$. Coosemans et al. (Reference Coosemans, Dekeyser and Baelmans2022) found that the three transport coefficient are proportional to one another up to a constant of order unity for the investigated 2-D, electrostatic, interchange-dominated, sheath-connected SOL cases. It remains to be confirmed if this scaling holds more generally. For now, it is assumed that this results holds to a certain extent in more complex cases as well. It stands to reason to propose a similar diffusion relation for the turbulent $E\times B$ flux of parallel momentum

(6.2)\begin{equation} \boldsymbol{\varGamma}_{n\boldsymbol{V}_{\|},E} \triangleq \overline{n\boldsymbol{V}_{E}''V_{\|}''}\approx-\nu\bar{n}\boldsymbol{\nabla}_{{\perp}}\tilde{V}_{\|}, \end{equation}

and to assume that $\nu \sim D$ as well.

Under these assumptions, it suffices to model only one of the transport coefficients to obtain a model for all four. Drawing on Coosemans et al. (Reference Coosemans, Dekeyser and Baelmans2020, Reference Coosemans, Dekeyser and Baelmans2021a,Reference Coosemans, Dekeyser and Baelmansb, Reference Coosemans, Dekeyser and Baelmans2022) and Coosemans (Reference Coosemans2022), an elaborate scaling of the form

(6.3)\begin{equation} D = C_D\frac{k_{E}}{\sqrt{m\zeta_{E}}+C_S mS_{m}}\sin(\varPsi) ,\end{equation}

may be suggested. The philosophy of this model is that $k_{E}$ provides a velocity scale for the turbulence, while the enstrophy $\zeta _{E}$ and the flow shear rate $S_{m}$ provide competing time scales for the turbulence. Finally, the effective phase difference between the density and potential fluctuations $\varPsi$ provides an additional correction. This phase difference is typically a characteristic of the turbulent regime, and could presumably be modelled based on which terms are (locally) dominant in the $k_{E}$ equation. It might still be necessary to adjust the coefficients $C_D$ and/or $C_S$ depending on the turbulence regime (open or closed field lines, interchange- or DW-dominated case, sheath connected or not). This transport coefficient model, how general it is and how far case-specific parameter tuning is needed, certainly require further investigation in realistic 3-D cases. However, given its rather robust performance in Coosemans et al. (Reference Coosemans, Dekeyser and Baelmans2021b, Reference Coosemans, Dekeyser and Baelmans2022), it seems reasonable to assume that the basic scaling

(6.4)\begin{equation} D \sim \rho\sqrt{k_{E}/m} ,\end{equation}

will hold to some extent, even though the proportionality factor may not take a strictly fixed value that works for all cases. Assuming that this scaling holds, the next section will summarise the progress that has been made in modelling the transport equation of $k_{E}$, and which terms and effects require further attention. Besides the relevance for the transport coefficients themselves, this also led to insights into the dynamics and drivers of the turbulence itself.

6.2. Modelling the turbulent kinetic energy equation

The general $E\times B$ turbulent kinetic energy equation for a low $\beta$ edge plasma has been derived in § 5.1 as (5.6). We start by simplifying this equation ignoring a number of terms presumed to be of minor importance

(6.5)\begin{align} & \frac{\partial \bar{n}k_{E}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\left(\bar{n}k_{E}\tilde{\boldsymbol{V}}_{C} + \frac{m\overline{n\boldsymbol{V}_{E}''^2\boldsymbol{V}_{C}''}}{2} + \overline{\phi'\boldsymbol{J}_{\|}'}\right) \nonumber\\ & \quad =\overline{J_{\|}'\nabla_{\|}\phi'} + \overline{p'\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{V}_{E}'} - m\overline{n\boldsymbol{V}_{C}''\boldsymbol{V}_{E}''}: \boldsymbol{\nabla}\tilde{\boldsymbol{V}}_{E}^{\rm T} + \overline{\boldsymbol{\nabla}\phi'\boldsymbol{\cdot}\boldsymbol{J}_{p,*}'} \nonumber\\ & \qquad - m\overline{nV_{\|}\frac{{\rm D}\boldsymbol{b}}{{\rm D} t}\boldsymbol{\cdot}\boldsymbol{V}_{E}''} - \frac{m\overline{S_{n_{i}}\boldsymbol{V}_{E}''^2}}{2} - m\tilde{\boldsymbol{V}}_{E}\boldsymbol{\cdot} \overline{\boldsymbol{V}_{E}''S_{n_{i}}} + \overline{\boldsymbol{S}_{m}\boldsymbol{\cdot}\boldsymbol{V}_{E}''}. \end{align}

Firstly, the sum of the transport terms $\boldsymbol {\nabla }\boldsymbol {\cdot }(\overline {\phi '\boldsymbol {J}_{\perp }'}+\overline {p'\boldsymbol {V}_{E}'})$ has been neglected. Even though each of these terms individually may be large, Scott (Reference Scott2003) argues that their sum is small due to Poynting cancellation. Moreover, under a low $\beta$ approximation the subterm $\boldsymbol {\nabla }\boldsymbol {\cdot }(\overline {\phi '\boldsymbol {J}_{*}'}+\overline {p'\boldsymbol {V}_{E}'})=\boldsymbol {\nabla }\boldsymbol {\cdot }(\boldsymbol {\nabla }(\overline {p'\phi '})\times \boldsymbol {b}/B)$ can be shown to depend solely on magnetic field gradients, of which the length scale is supposed to be much larger than the turbulent length scale (Scott Reference Scott2003; Coosemans et al. Reference Coosemans, Dekeyser and Baelmans2021b). Next, the terms involving the viscous stresses have likewise been neglected. On evaluations of this $k_{E}$ equation for simplified TOKAM2D cases, these terms were observed to always play a secondary role (Coosemans et al. Reference Coosemans, Dekeyser and Baelmans2020, Reference Coosemans, Dekeyser and Baelmans2021b, Reference Coosemans, Dekeyser and Baelmans2022). This complies with the general assumption that plasma edge turbulence is dominated by an inviscid dynamics (Camargo et al. Reference Camargo, Biskamp and Scott1995). Lastly, the Favre term has likewise been left away since all earlier TOKAM2D results showed it to be small as well.

The interchange term (second term on the right-hand side of (6.5)) is widely accepted to act like the major drive of the turbulence in the SOL on the low field side (LFS) in many cases. A general analytical equation has been derived for this term by Coosemans et al. (Reference Coosemans, Dekeyser and Baelmans2021b, Reference Coosemans, Dekeyser and Baelmans2022). This starts by rewriting (Scott Reference Scott2003)

(6.6)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{V}_{E} =- \boldsymbol{V}_{E}\boldsymbol{\cdot}\boldsymbol{\nabla}\ln{B^2} + \frac{\boldsymbol{\nabla}\phi}{B^2}\boldsymbol{\cdot}\boldsymbol{\nabla} \times B,\end{equation}

where the second term can be neglected by a low-$\beta$ approximation. Rewriting the fluctuations and using the definitions of the $E\times B$ particle and heat fluxes in (3.15a–c) and (3.15a–c), it is found that (Coosemans et al. Reference Coosemans, Dekeyser and Baelmans2022)

(6.7)\begin{align} \overline{p'\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{V}_{E}'} & =- \overline{p'\boldsymbol{V}_{E}'}\boldsymbol{\cdot}\boldsymbol{\nabla}\ln{B^2} =- (\tilde{T}\overline{n'\boldsymbol{V}_{E}'} + \overline{nT''\boldsymbol{V}_{E}''})\boldsymbol{\cdot}\boldsymbol{\nabla}\ln(B^2) \nonumber\\ & =-(\tilde{T}\boldsymbol{\varGamma}_{n,t,E} + \boldsymbol{\varGamma}_{p_{{i}},t,E} + \boldsymbol{\varGamma}_{p_{{e}},t,E})\boldsymbol{\cdot}\boldsymbol{\nabla}\ln(B^2). \end{align}

Hence, if closure models are available for the $E\times B$ fluxes, whether of the form suggested in § 6.1 or not, the interchange source of $k_{E}$ can be evaluated analytically. This provides a very convenient ingredient for a closure model. Moreover, it also automatically induces a ballooning character into a turbulent transport model as already remarked in earlier publications (Coosemans et al. Reference Coosemans, Dekeyser and Baelmans2020, Reference Coosemans, Dekeyser and Baelmans2021b, Reference Coosemans, Dekeyser and Baelmans2022). Indeed, the turbulent heat fluxes will normally be outward everywhere, while the magnetic field gradient is only opposed to this flux on the LFS. As a result, the interchange term tends to act like a source on the LFS, while it acts like a sink on the high field side (HFS). This behaviour has been confirmed in 2-D mean-field simulations with SOLPS-ITER in which a model based on the $k_{\perp }$ equation presented here has been implemented (Dekeyser et al. Reference Dekeyser, Coosemans, Carli and Baelmans2022). Note that this interchange model holds in general and is thus not only relevant in interchange-dominated turbulence regimes. It is interesting to remark that, even when another term would provide a more poloidally uniform drive of $k_{\perp }$, the turbulence on the HFS would be decreased and that on the LFS increased due to the resulting turbulent $E\times B$ heat fluxes.

In sheath-connected interchange-dominated LFS SOL cases studied with the TOKAM2D code, it was found that this interchange term is balanced by the parallel loss to the sheath (Coosemans et al. Reference Coosemans, Dekeyser and Baelmans2020, Reference Coosemans, Dekeyser and Baelmans2021b, Reference Coosemans, Dekeyser and Baelmans2022). This term is the equivalent of the transport part of the parallel current contribution (fourth term on the left-hand side of (6.5)). It was also found that the underlying ‘anomalous’ parallel flux $\overline {\phi '\boldsymbol {J}_{\|}'}$ largely exceeds the equivalent of the mean-field and the turbulent parallel convection.

This has interesting implications for the distribution of $k_{\perp }$ in tokamaks. The exploratory SOLPS-ITER case presented by Dekeyser et al. (Reference Dekeyser, Coosemans, Carli and Baelmans2022) had the interchange term as the dominant source of $k_{\perp }$ with this fast transport acting like the main compensation mechanism. The resulting balance showed strong production of turbulence around the outer midplane (OMP), which is quickly removed by the fast parallel transport. In the SOL, this allows saturation of the turbulence by removal of $k_{\perp }$ to the divertor targets, which is reminiscent of the physics of the TOKAM2D cases. In the core region, on the other hand, $k_{\perp }$ can be transported away from the OMP to the HFS, where it may be dissipated by the interchange term acting like a sink in that region. Hence, this term adds a strongly non-local element to the balance of $k_{\perp }$ and thus to the turbulent transport of particles and heat. Furthermore, this increased parallel spreading of $k_{\perp }$ also means that the resulting turbulent transport is less ballooned than would have been the case had the interchange source (on the LFS) been balanced by a local sink. More research, however, is needed to develop a reliable model for this anomalous parallel flux.

It is not claimed in this article that this flow picture or the corresponding saturation mechanism is generally valid in any tokamak operating regime. Nonetheless, the insights provided by them and the role of the $\overline {\phi '\boldsymbol {J}_{\|}'}$ flux in enhancing the parallel spreading of the turbulence are believed to be novel and may in general be considered in the overall flow picture in tokamaks. Many other effects which were not yet taken into account may, however, modify this dynamics.

6.3. Limitations of the $k_{\perp }$ model

In the current stage of its development, the validity of the $k_{\perp }$ model as presented by Coosemans et al. (Reference Coosemans, Dekeyser and Baelmans2022) and Dekeyser et al. (Reference Dekeyser, Coosemans, Carli and Baelmans2022) is limited since it has mostly been developed and tested for 2-D, interchange-dominated, sheath-connected, electrostatic SOL cases. A number of terms which are expected to be important in the $k_{E}$ transport equation (6.5) in different cases were neglected.

The first and third terms in (6.5) associated with the DW and RS energy transfer channel (see § 4.4) have for example not yet been duly modelled, even though they are expected to be of primary importance for many machine-relevant cases. The exact form of these terms in the equation for $k_{E}$ in particular allows their further analysis in the context of the development of an improved $k_{\perp }$ model. From a preliminary analysis for a limited number of cases in Coosemans (Reference Coosemans2022), it seems that the DW term can change significantly depending on the regime the turbulence is in and how exactly the competition with other terms occurs. Much more analysis is thus required to model this term.

The RS term is expected to be of minor importance in the SOL because the flow shear is limited by the electrostatic potential being tied to the sheath potential at the end of the field lines. However, around the separatrix and in the core region, strong poloidal flows, which are sheared in the radial direction, may develop, such that the $m\overline {nV_{C,r}'V_{E,\theta }''}\partial _r\tilde {V}_{E,\theta }$ contribution in particular may become important. Primarily, this term is expected to break up the turbulent eddies and as such act like a sink of $k_{\perp }$ (in line with the inverse energy cascade Yakhot Reference Yakhot2004; Alexakis & Doering Reference Alexakis and Doering2006; Fundamenski Reference Fundamenski2010). However, when the turbulence is sufficiently damped and when the flow shear is sufficiently strong, the KH instability may come into play. This secondary instability might then lead the RS term to act like a source of $k_{\perp }$ (Rogers & Dorland Reference Rogers and Dorland2005; Myra et al. Reference Myra, D'Ippolito, Russell, Umansky and Baver2016; Giacomin & Ricci Reference Giacomin and Ricci2020). Furthermore, as indicated in (6.3), strong flow shear could also influence the diffusion relation itself. Hence, flow shear is expected to lead to both a reduction of $k_{E}$ and a reduction of the transport coefficients at the same $k_{E}$. In addition, since both effects depend on the flow shear, a model for $\partial _r\tilde {V}_{E,\theta }$ itself is required. While the mean-field $E\times B$ velocity can be calculated from the mean-field charge balance equation (3.12) for the electrostatic potential, the closure terms in this equation need further attention.

Furthermore, the effects of perpendicular turbulent transport of $k_{E}$ might also require further attention. In the TOKAM2D cases studied earlier, its contribution to the balance of $k_{\perp }$ was found to be small. However, this may not be the case in general (Ghendrih et al. Reference Ghendrih, Sarazin, Ciraolo, Darmet, Garbet, Grangirard, Tamain, Benkadda and Beyer2007). In regions where the perpendicular gradients of $k_{\perp }$ are steeper, more radial transport of it would likewise be expected (Singh & Diamond Reference Singh and Diamond2020). Such strong gradients are expected around the separatrix due to strong flow shear, and perhaps also in the divertor at the boundary between the private flux region and the SOL.

The phenomena included in the last five terms in (6.5) have been subject to far less study as far as the authors are aware. This makes it interesting to first evaluate these terms with detailed reference data, from turbulence code simulations for example. This way, either it could be found that they have rightfully not been given much consideration, or it might open the door to valuable new insights. In a next step, the relevant terms could then be modelled and incorporated in an extended $k_{\perp }$ model.

The last three terms in (6.5) arise through (fluctuating) particle and momentum source terms. These source terms are expected to be mainly caused by plasma–neutral interactions. While these interactions have long been considered in mean-field transport modelling (Reiter Reference Reiter1992; Rognlien et al. Reference Rognlien, Ryutov, Mattor and Porter1999; Wiesen Reference Wiesen2006; Dekeyser Reference Dekeyser2014; Dekeyser et al. Reference Dekeyser, Reiter and Baelmans2014; Bufferand et al. Reference Bufferand, Ciraolo, Marandet, Bucalossi, Ghendrih, Gunn, Mellet, Tamain, Leybros and Fedorczak2015; Wiesen et al. Reference Wiesen, Reiter, Kotov, Baelmans, Dekeyser, Kukushkin, Lisgo, Pitts, Rozhansky and Saibene2015; Bonnin et al. Reference Bonnin, Dekeyser, Pitts, Coster, Voskoboynikov and Wiesen2016; Simonini et al. Reference Simonini, Corrigan, Radford, Spence and Taroni2018), their study in the context of edge turbulence and their integration in turbulence codes has only been picked up more recently (Marandet et al. Reference Marandet, Tamain, Futtersack, Ghendrih, Bufferand, Genesio and Mekkaoui2013; Wersal & Ricci Reference Wersal and Ricci2015; Thrysøea et al. Reference Thrysøea, Løiten, Madsen, Naulin, Nielsen and Rasmussen2018; Fan et al. Reference Fan, Marandet, Tamain, Bufferand, Ciraolo, Ghendrih and Serre2019; Bufferand et al. Reference Bufferand, Bucalossi, Ciraolo, Falchetto, Gallo, Ghendrih, Rivals, Tamain, Yang and Giorgiani2021; Zholobenko et al. Reference Zholobenko, Stegmeir, Griener, Conway, Body, Coster and Jenko2021; Giacomin et al. Reference Giacomin, Ricci, Coroado, Fourestey, Galassi, Lanti, Mancini, Richart, Stenger and Varini2022). While the recent results obtained show the neutrals to have an important influence on the turbulence, it remains to be established if the source terms identified in (6.5) are large (in certain regimes and regions), or whether their effect is more indirect.

Likewise, the literature has already found that including the contribution of the diamagnetic drift to the polarisation current changes the dynamics of the turbulence (Madsen et al. Reference Madsen, Garcia, Larsen, Naulin, Nielsen and Rasmussen2011; Bisai & Kaw Reference Bisai and Kaw2013; Baudoin Reference Baudoin2018). This contribution is taken into account in most modern turbulence codes, although Coosemans (Reference Coosemans2022) showed that care should be taken to not introduce unphysical terms when implementing this. However, this effect has not yet been duly studied in the context of source terms for $k_{E}$ (fourth term on the right-hand side of (6.5)). Again, further research should investigate if this term is large, or whether the influence of the inertia of the diamagnetic drift is through other (indirect) mechanisms. For this, the diamagnetic and mixed kinetic energy equations (5.11) and (5.16) might likewise be studied.

Finally, the $\textrm {D}\boldsymbol {b}/\textrm {D} t$ term is commonly neglected in the polarisation current defined in (2.11) which enters in the charge balance equation (2.20) and in the parallel ion momentum equation (2.23). This is also the case in modern turbulence codes. As such, the energetic coupling between the parallel and perpendicular kinetic energy and the fifth term on the right-hand side of (6.5) is not taken into account in these codes. Being the only direct transfer channel between parallel and perpendicular kinetic energy and in particular coupling $k_{E}$ with the mean-field parallel flows which are known to be large, it might be speculated to provide, however, a significant source of energy to the turbulence. Evaluating the $\textrm {D}\boldsymbol {b}/\textrm {D} t$ term in the $k_{E}$ equation (a posteriori) would allow to verify this assumption. If it were found to be important, it could allow to improve the accuracy of the turbulence codes. Later on, it could then be further studied and modelled for use in an extended $k_{\perp }$ model as well.

Depending on which of the terms in the $k_{\perp }$ equation are dominant in a certain case, i.e. which regime the turbulence is in, it can be expected that the structure of the turbulence is different. As a result, it is likely that the exact relation between the intensity of the turbulence (e.g. $k_{\perp }$) and the transport resulting from it (i.e. the turbulent transport coefficients) will be different. Hence, it is possible that depending on the regime, different model constants might be needed in (6.4) or even a different form of the transport relation itself. The occurrence of different turbulence regimes and the threshold values for the transition between them have been studied in the literature, see for example Scott (Reference Scott2005), Mosetto et al. (Reference Mosetto, Halpern, Jolliet, Loizu and Ricci2013), Giacomin & Ricci (Reference Giacomin and Ricci2020) and Eich et al. (Reference Eich and Manz2021).

Furthermore, models that are not purely diffusive for the turbulent fluxes could be of interest since the literature indicates that particle and heat transport in the plasma edge is in fact driven by radially propagating structures such as avalanches and blob filaments. They would rather induce intermittent convective/ballistic transport with a strong non-local character (Politzer et al. Reference Politzer, Austin, Gilmore, McKee, Rhodes, Yu, Doyle, Evans and Moyere2002; Garcia et al. Reference Garcia, Naulin, Nielsen and Rasmussen2006; Naulin Reference Naulin2007; Krasheninnikov, D'Ippolito & Myra Reference Krasheninnikov, D'Ippolito and Myra2008; Ghendrih et al. Reference Ghendrih, Ciraolo, Larmande, Sarazin, Tamain, Beyer, Chiavassa, Darmet, Garbet and Grandgirard2009; D'Ippolito, Myra & Zweben Reference D'Ippolito, Myra and Zweben2011). It might, however, be expected that a well-chosen diffusion model can give a reasonable approximation of the long time scale average particle flux caused by all the instantaneous filaments with high and low density structures moving respectively outward and inward in a seemingly random way. Note also that the models for the transport coefficients do not need to use local quantities only. The transport of $k_{E}$ introduces a non-local effect in the mean-field model for example. This transport allows turbulent kinetic energy created in one location to increase turbulent transport in another.

Hence, it is expected that a basic scaling between the transport coefficients and well-chosen quantifiers of the turbulence can be used to get an idea of the characteristics of the transport. It has indeed been proven in earlier work (Coosemans et al. Reference Coosemans, Dekeyser and Baelmans2020, Reference Coosemans, Dekeyser and Baelmans2021b, Reference Coosemans, Dekeyser and Baelmans2022) that such models can provide very accurate predictions for particular regimes such as electrostatic interchange-dominated $E\times B$ turbulence in a sheath-connected SOL.