3.1. A brief explanation of the topological velocity and non-barotropic cross-helicity

The mathematical expression for the cross-helicity of non-barotropic fluids is given by (Webb et al. Reference Webb, Dasgupta, McKenzie, Hu and Zank2014a; Yahalom Reference Yahalom2017a,Reference Yahalomb)

(3.1)\begin{equation} H_{{\rm CNB}} =\int {\rm d}^3x\, \boldsymbol{v}_t\boldsymbol{\cdot}\boldsymbol{B}. \end{equation}

Here, the topological velocity field is defined as $\boldsymbol {v_t}=\boldsymbol {v}-\sigma \boldsymbol {\nabla }s$ (Yahalom & Qin Reference Yahalom and Qin2021), where $\sigma$ is an auxiliary variable, which depends on the Lagrangian time integral of the temperature i.e.

(3.2)\begin{equation} \frac{{\rm d}\sigma}{{\rm d}t}=T. \quad \left(\Rightarrow \sigma = \int_{0}^{t} T\,{\rm d}t'+\sigma_0 = t \langle T\rangle +\sigma_0\right). \end{equation}

The above integral is done for each fluid element separately and $\langle \rangle =({1}/{t} )\int _{0}^{t} \,{\rm d}t'$ designates temporal averaging. In Appendix A we describe the variational approach leading to (3.2).

Notice that, in non-barotropic MHD, one can calculate the temporal derivative of the cross-helicity equation (1.1) using the standard equations to obtain

(3.3)\begin{equation} \frac{{\rm d}H_C}{{\rm d}t}=\int T\boldsymbol{\nabla}s\boldsymbol{\cdot}\boldsymbol{B}\,{\rm d}^3 x . \end{equation}

Thus, generically, cross-helicity is not conserved. A clue on how to define the cross-helicity for non-barotropic MHD can be obtained from the variational analysis described in Yahalom (Reference Yahalom2016b), which is valid for magnetic field lines at the intersection of two comoving surfaces $\chi$, $\eta _0$ (Euler potentials). Following Sakurai (Reference Sakurai1979), the magnetic field takes the form

(3.4)\begin{equation} \boldsymbol{B}=\boldsymbol{\nabla}\chi\times\boldsymbol{\nabla}\eta_0. \end{equation}

And the generalized Clebsch representation of the velocity (Yahalom Reference Yahalom2016b) is

(3.5)\begin{equation} \boldsymbol{v}=\boldsymbol{\nabla}\nu+\alpha\boldsymbol{\nabla}\chi+\beta\boldsymbol{\nabla} \eta_0+\sigma\boldsymbol{\nabla}s, \end{equation}

where, in the above, $\alpha$, $\beta$ and $\nu$ are Lagrange multipliers appearing in the said action (Yahalom Reference Yahalom2016b); see also Appendix A. Let us now write the cross-helicity given in (1.1) in terms of (3.4) and (3.5), this will take the form

(3.6)\begin{equation} H_C=\int {\rm d}\varPhi[\nu]+\int {\rm d}\varPhi\oint\sigma \,{\rm d}s, \end{equation}

in which ${\rm d}\varPhi = \boldsymbol {B} \boldsymbol {\cdot } {\rm d} \boldsymbol {S}$ is the magnetic flux, and $[\nu ]$ is the discontinuity of the non-single valued potential $\nu$ (Yahalom Reference Yahalom2017a). Now, as for ideal MHD, the magnetic field lines move with the flow, and it follows that the magnetic flux ${\rm d}\varPhi$ is conserved. It is also shown in Yahalom (Reference Yahalom2017a) that the material derivative of $[\nu ]$ must vanish. Thus, the first term on the right-hand side of (3.6) is conserved. This suggests the following definition for the non-barotropic cross-helicity $H_{{\rm CNB}}$

(3.7)\begin{equation} H_{{\rm CNB}}=\int {\rm d}\varPhi[\nu]=H_C-\int {\rm d}\varPhi\oint\sigma \,{\rm d}s. \end{equation}

The conventional form of the same expression is given in (3.1). Please refer to Yahalom (Reference Yahalom2017a) for the detailed justification of the definition, the form of the non-barotropic cross-helicity and a proof of its constancy.

We point out that, from a pure mathematical point of view, it does not really matter what the physical meaning of the vector fields appearing in the cross-helicity term is, the cross-helicity will describe how the fields are knotted together (Yokoi Reference Yokoi2013, § 2, see also Ricca & Moffatt (Reference Ricca and Moffatt1992) and Ricca & Berger (Reference Ricca and Berger1996) for non-crossed helicity). Thus, from a purely topological point of view, it does not matter if we consider the physical velocity field $\boldsymbol {v}$ or the topological velocity field $\boldsymbol {v}_t$.

3.2. The temporal derivative of non-barotropic cross-helicity

Next, we study the temporal derivative of the non-barotropic cross-helicity

(3.8)\begin{equation} \frac{{\rm d} H_{{\rm CNB}}}{{\rm d}t} =\int {\rm d}^3 x\left(\boldsymbol{v}_t \boldsymbol{\cdot} \frac {\partial \boldsymbol{B}}{\partial t} +\boldsymbol{B} \boldsymbol{\cdot} \frac {\partial \boldsymbol{v}_t}{\partial t}\right). \end{equation}

Now we calculate the first term on the right-hand side with the help of (2.4)

(3.9)\begin{gather} \boldsymbol{v}_t \boldsymbol{\cdot}\frac{\partial \boldsymbol{B}}{\partial t}=\boldsymbol{v}_t\boldsymbol{\cdot} \left\{\boldsymbol{\nabla} \times (\boldsymbol{v}\times\boldsymbol{B}) +\frac{\eta}{4{\rm \pi}}\nabla^2\boldsymbol{B}\right\} \end{gather}

(3.10)\begin{gather}\boldsymbol{v}_t \boldsymbol{\cdot}\frac{\partial \boldsymbol{B}}{\partial t}=\boldsymbol{\nabla}\boldsymbol{\cdot}\{(\boldsymbol{v}\times\boldsymbol{B})\times\boldsymbol{v}_t\}+(\boldsymbol{v}\times\boldsymbol{B})\boldsymbol{\cdot}\boldsymbol{\omega}_t+\boldsymbol{v}_t\boldsymbol{\cdot}\frac{\eta}{4{\rm \pi}}\nabla^2\boldsymbol{B}, \end{gather}

where we define the topological vorticity of the topological flow field as

(3.11)\begin{equation} \boldsymbol{\omega}_t \equiv \boldsymbol{\nabla}\times \boldsymbol{v}_t. \end{equation}

Next, we calculate the second term on the right-hand side

(3.12)\begin{equation} \boldsymbol{B}\boldsymbol{\cdot}\frac {\partial \boldsymbol{v}_t}{\partial t}= \boldsymbol{B}\boldsymbol{\cdot}\frac {\partial (\boldsymbol{v}-\sigma\boldsymbol{\nabla} s)}{\partial t}= \boldsymbol{B}\boldsymbol{\cdot}\left(\frac{\partial \boldsymbol{v}}{\partial t}-\frac{\partial \sigma}{\partial t}\boldsymbol{\nabla} s-\sigma\boldsymbol{\nabla}\frac{\partial s}{\partial t}\right). \end{equation}

Now we simplify the right-hand side of (3.12) in three steps: the first term is calculated with the help of (2.1)

(3.13)\begin{align} \frac{\partial \boldsymbol{v}}{\partial t}& ={-}(\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{v}-\frac{\boldsymbol{\nabla} p}{\rho}+\frac{(\boldsymbol{\nabla}\times\ \boldsymbol{B})\times \boldsymbol{B}}{4{\rm \pi}\rho}-\boldsymbol{\nabla}\phi+\frac{1}{\rho}\frac{\partial \sigma_{ik}^\prime}{\partial x_k} \nonumber\\ & =(\boldsymbol{v}\times\boldsymbol{\omega})+\frac{(\boldsymbol{\nabla}\times \boldsymbol{B})\times \boldsymbol{B}}{4{\rm \pi}\rho}-\boldsymbol{\nabla}\left(\frac{v^2}{2}\right)-\boldsymbol{\nabla} w+T\boldsymbol{\nabla} s-\boldsymbol{\nabla}\phi+\frac{1}{\rho}\frac{\partial \sigma_{ik}^\prime}{\partial x_k} , \end{align}

in which the vorticity is

(3.14)\begin{equation} \boldsymbol{\omega} \equiv \boldsymbol{\nabla}\times \boldsymbol{v}, \end{equation}

and we have used the thermodynamic identity

(3.15)\begin{equation} {\rm d}w = {\rm d}\varepsilon + {\rm d}\left(\frac{p}{\rho}\right) = T\,{\rm d}s + \frac{1}{\rho} {\rm d}p \Rightarrow \boldsymbol{\nabla} w = T \boldsymbol{\nabla} s + \frac{1}{\rho} \boldsymbol{\nabla} p. \end{equation}

Thus

(3.16)\begin{equation} \boldsymbol{B}\boldsymbol{\cdot}\frac{\partial \boldsymbol{v}}{\partial t}= \boldsymbol{B}\boldsymbol{\cdot}\left\{(\boldsymbol{v}\times\boldsymbol{\omega})- \boldsymbol{\nabla}\left(\frac{v^2}{2}+w+\phi\right)+T\boldsymbol{\nabla} s\right\}+\frac{B_i}{\rho}\frac{\partial \sigma_{ik}^\prime}{\partial x_k}. \end{equation}

In the second term, we use (3.2) to obtain

(3.17)\begin{equation} {-}\frac{\partial\sigma}{\partial t}\boldsymbol{\nabla} s=(\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla}\sigma-T)\boldsymbol{\nabla} s. \end{equation}

In the third term, we use (2.5) to derive

(3.18)\begin{equation} {-}\sigma\boldsymbol{\nabla}\frac{\partial s}{\partial t}=\sigma\boldsymbol{\nabla}\left[\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla}s-\frac{1}{\rho T}\sigma_{ik}^\prime\frac{\partial v_{i}}{\partial x_{k}}-\frac{\eta}{\rho T}J^2-\frac{1}{\rho T}\boldsymbol{\nabla}\boldsymbol{\cdot}(k\boldsymbol{\nabla}T)\right] . \end{equation}

Combining the above expressions, we have

(3.19)\begin{align} \boldsymbol{B}\boldsymbol{\cdot}\frac{\partial \boldsymbol{v}_t}{\partial t}& = \boldsymbol{B}\boldsymbol{\cdot}\left[(\boldsymbol{v}\times\boldsymbol{\omega})- \boldsymbol{\nabla}\left(\frac{v^2}{2}+w +\phi\right)+(\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla}\sigma) \boldsymbol{\nabla} s+\sigma\boldsymbol{\nabla}(\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla} s) \right. \nonumber\\ & \quad + \left. \sigma\boldsymbol{\nabla}\left\{-\frac{1}{\rho T}\sigma_{ik}^\prime\frac{\partial v_{i}}{\partial x_{k}}- \frac{\eta}{\rho T}J^2-\frac{1}{\rho T}\boldsymbol{\nabla}\boldsymbol{\cdot}( k\boldsymbol{\nabla}T)\right\}\right]+\frac{B_i}{\rho}\frac{\partial \sigma_{ik}^\prime}{\partial x_k}. \end{align}

Notice that

(3.20)\begin{equation} \boldsymbol{\nabla} \{\sigma(\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla} s)\} = \sigma \boldsymbol{\nabla} (\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla} s)+ (\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla} s ) \boldsymbol{\nabla} \sigma, \end{equation}

and also that

(3.21)\begin{align} \boldsymbol{v} \times \boldsymbol{\omega}_t & = \boldsymbol{v} \times (\boldsymbol{\omega} - \boldsymbol{\nabla} \sigma \times \boldsymbol{\nabla} s)= \boldsymbol{v} \times \boldsymbol{\omega} + (\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \sigma ) \boldsymbol{\nabla} s - (\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} s ) \boldsymbol{\nabla} \sigma \nonumber\\ & = \boldsymbol{v} \times \boldsymbol{\omega} + (\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \sigma ) \boldsymbol{\nabla} s - \boldsymbol{\nabla} ((\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} s ) \sigma ) +\sigma \boldsymbol{\nabla} (\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} s ) , \end{align}

or that

(3.22)\begin{equation} \boldsymbol{v} \times \boldsymbol{\omega}_t+\boldsymbol{\nabla} ((\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} s ) \sigma ) = \boldsymbol{v} \times \boldsymbol{\omega} + (\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \sigma ) \boldsymbol{\nabla} s +\sigma \boldsymbol{\nabla} (\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} s ) . \end{equation}

Thus we obtain

(3.23)\begin{align} \boldsymbol{B}\boldsymbol{\cdot}\frac{\partial\boldsymbol{v}_t}{\partial\ t}& = \boldsymbol{B}\boldsymbol{\cdot}\left[(\boldsymbol{v}\times\boldsymbol{\omega}_t)+\boldsymbol{\nabla} \left\{\sigma(\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla} s)-\frac{v^2}{2}-w-\phi\right\}\right] \nonumber\\ & \quad -\boldsymbol{B}\boldsymbol{\cdot}\left[\sigma\boldsymbol{\nabla}\left\{\frac{1}{\rho T}\sigma_{ik}^\prime\frac{\partial v_{i}}{\partial x_{k}}+\frac{\eta}{\rho T}J^2+\frac{1}{\rho T}\boldsymbol{\nabla}\boldsymbol{\cdot}(k\boldsymbol{\nabla}T)\right\}\right]+ \frac{B_i}{\rho}\frac{\partial \sigma_{ik}^\prime}{\partial x_k} . \end{align}

Combining (3.10) and (3.23) and taking into account that

(3.24)\begin{equation} \boldsymbol{B}\boldsymbol{\cdot} (\boldsymbol{v}\times\boldsymbol{\omega}_t) ={-} (\boldsymbol{v}\times \boldsymbol{B}) \boldsymbol{\cdot} \boldsymbol{\omega}_t ,\end{equation}

we obtain

(3.25)\begin{align} \boldsymbol{v}_t\boldsymbol{\cdot}\frac{\partial \boldsymbol{B}}{\partial t}+\frac{\partial \boldsymbol{v}_t}{\partial t}\boldsymbol{\cdot} \boldsymbol{B}& =\boldsymbol{\nabla}\boldsymbol{\cdot}\{(\boldsymbol{v}\times \boldsymbol{B})\times \boldsymbol{v}_t\}+\frac{\eta}{4{\rm \pi}}\boldsymbol{v}_t\boldsymbol{\cdot}\nabla^2 \boldsymbol{B}\nonumber\\ & \quad +\boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla}\left\{\sigma(\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla} s)-\frac{v^2}{2}-w-\phi\right\} \nonumber\\ & \quad +\frac{B_i}{\rho}\frac{\partial \sigma_{ik}^\prime}{\partial x_k}- \sigma\boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla}\left\{\frac{1}{\rho T}\sigma_{ik}^\prime\frac{\partial v_{i}}{\partial x_{k}}+\frac{\eta}{\rho T}J^2+\frac{k}{\rho T}\nabla^2T \right\} \nonumber\\ & = \boldsymbol{\nabla} \boldsymbol{\cdot} \left[\{(\boldsymbol{v}\times \boldsymbol{B})\times \boldsymbol{v}_t\} +\boldsymbol{B}\left\{\sigma(\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla}s) -\frac{v^2}{2}-w-\phi \right\}\right] \nonumber\\ & \quad - \left. \sigma \boldsymbol{B} \left\{\frac{1}{\rho T}\sigma_{ik}^\prime \frac{\partial v_{i}}{\partial x_{k}}+\frac{\eta}{\rho T}J^2+\frac{k}{\rho T}\nabla^2T \right\} \right] \nonumber\\ & \quad +\frac{\eta}{4{\rm \pi}}\boldsymbol{v}_t\boldsymbol{\cdot}\nabla^2 \boldsymbol{B}+\frac{B_i}{\rho}\frac{\partial \sigma_{ik}^\prime}{\partial x_k} \nonumber\\ & \quad + (\boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{\nabla} \sigma) \left[ \frac{1}{\rho T}\sigma_{ik}^\prime\frac{\partial v_{i}}{\partial x_{k}}+\frac{\eta}{\rho T}J^2+\frac{k}{\rho T}\nabla^2T \right] . \end{align}

Now, substituting (3.25) into (3.8), we obtain

(3.26)\begin{align} \frac{{\rm d}H_{{\rm CNB}}}{{\rm d}t}& =\int \boldsymbol{\nabla} \boldsymbol{\cdot} \left[\{(\boldsymbol{v}\times \boldsymbol{B})\times \boldsymbol{v}_t\} +\boldsymbol{B}\left\{\sigma(\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla}s)-\frac{v^2}{2}-w-\phi \right\} \right. \nonumber\\ & \quad-\left. \sigma \boldsymbol{B} \left\{\frac{1}{\rho T}\sigma_{ik}^\prime\frac{\partial v_{i}}{\partial x_{k}}+\frac{\eta}{\rho T}J^2+\frac{k}{\rho T}\nabla^2T \right\}\right]{\rm d}^3x \nonumber\\ & \quad + \int \left\{ \frac{\eta}{4{\rm \pi}}\boldsymbol{v}_t\boldsymbol{\cdot}\nabla^2 \boldsymbol{B}+\frac{B_i}{\rho}\frac{\partial \sigma_{ik}^\prime}{\partial x_k} \right. \nonumber\\ & \quad +\left. (\boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{\nabla} \sigma) \left[ \frac{1}{\rho T}\sigma_{ik}^\prime\frac{\partial v_{i}}{\partial x_{k}}+\frac{\eta}{\rho T}J^2+\frac{k}{\rho T}\nabla^2T \right]\right\}{\rm d}^3x , \end{align}

using Gauss’ divergence theorem, we obtain

(3.27)\begin{align} \frac{{\rm d}H_{{\rm CNB}}}{{\rm d}t}& =\oint \left[\{(\boldsymbol{v}\times \boldsymbol{B})\times \boldsymbol{v}_t\} +\boldsymbol{B}\left\{\sigma(\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla}s)-\frac{v^2}{2}-w-\phi \right\} \right. \nonumber\\ & \quad -\left. \sigma \boldsymbol{B} \left\{\frac{1}{\rho T}\sigma_{ik}^\prime\frac{\partial v_{i}}{\partial x_{k}}+\frac{\eta}{\rho T}J^2+\frac{k}{\rho T}\nabla^2T\right\} \right]\boldsymbol{\cdot} {\rm d} \boldsymbol{S} \nonumber\\ & \quad + \int \left\{\frac{\eta}{4{\rm \pi}}\boldsymbol{v}_t\boldsymbol{\cdot}\nabla^2 \boldsymbol{B}+ \frac{B_i}{\rho}\frac{\partial \sigma_{ik}^\prime}{\partial x_k} \right. \nonumber\\ & \quad +\left.(\boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{\nabla} \sigma) \left[ \frac{1}{\rho T}\sigma_{ik}^\prime\frac{\partial v_{i}}{\partial x_{k}}+\frac{\eta}{\rho T}J^2+\frac{k}{\rho T}\nabla^2T \right]\right\}{\rm d}^3x . \end{align}

Here, the surface integral encapsulates the volume for which the cross-helicity is calculated. For generic problems arising in the solar and dynamo contexts, the volume chosen is not infinite. Experience with magnetic helicity variation shows that the boundary terms are often critical (and the largest in the solar corona), and the topology flowing in and out of systems is crucial in astrophysical contexts. It would likely be so for the non-barotropic cross-helicity. If the surface is taken at infinity the magnetic fields vanish and thus, in a generic case, the entire surface term, then the time derivative of cross-helicity can be written as

(3.28)\begin{equation} \frac{{\rm d} H_{{\rm CNB}}}{{\rm d}t}=\int\left\{ \frac{\eta}{4{\rm \pi}}\boldsymbol{v}_t\boldsymbol{\cdot}\nabla^2 \boldsymbol{B}+\frac{B_i}{\rho}\frac{\partial \sigma_{ik}^\prime}{\partial x_k} + \frac{\left(\boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{\nabla} \sigma\right)}{\rho T} \left(\sigma_{ik}^\prime\frac{\partial v_{i}}{\partial x_{k}}+\eta J^2+k \nabla^2T \right)\right\}{\rm d}^3x .\end{equation}

Thus, the time derivative of the cross-helicity depends generically on the stress tensor i.e. the viscosity of the fluid and the coefficient of magnetic diffusivity, and also on the heat conduction but not on heat convection. By putting all non-ideal terms to zero, we obtain the ideal MHD condition and conservation of non-barotropic cross-helicity takes place. In the special case that the magnetic field lies on $\sigma$ surfaces (that is, average temperature surfaces) and thus is orthogonal to $\boldsymbol {\nabla } \sigma$, the cross-helicity change will not depend on the thermal conductivity

(3.29)\begin{equation} \frac{{\rm d}H_{{\rm CNB}}}{{\rm d}t}=\int\left\{ \frac{\eta}{4{\rm \pi}}\boldsymbol{v}_t\boldsymbol{\cdot}\nabla^2 \boldsymbol{B}+\frac{B_i}{\rho}\frac{\partial \sigma_{ik}^\prime}{\partial x_k} \right\}{\rm d}^3x . \end{equation}

The same will be true for a high density and high temperature plasma, and a plasma of small temperature gradients (plasma in global thermal equilibrium). Of course, even if heat conduction does not affect the non-barotropic cross-helicity, other non-ideal processes do, and those include friction and ohmic losses.

To conclude this subsection we shall partition the time derivative of the non-barotropic cross-helicity in accordance with the non-ideal process that contributes to its modification

(3.30)\begin{align} \frac{{\rm d}H_{{\rm CNB}}}{{\rm d}t}& =\int\left\{ \eta \left[ \frac{\boldsymbol{v}_t \boldsymbol{\cdot} \nabla^2 \boldsymbol{B}}{4{\rm \pi}} + \frac{J^2}{\rho T} (\boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{\nabla}\sigma) \right] \right. \nonumber\\ & \quad + k (\boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{\nabla} \sigma)\frac{\nabla^2T}{\rho T} + \left. \frac{B_i}{\rho}\frac{\partial \sigma_{ik}^\prime}{\partial x_k} + (\boldsymbol{B} \boldsymbol{\cdot} \boldsymbol{\nabla} \sigma) \frac{\sigma_{ik}^\prime}{\rho T}\frac{\partial v_{i}}{\partial x_{k}}\right\}{\rm d}^3x . \end{align}

Let us introduce the dimensionless Reynolds number and magnetic Reynolds number

(3.31a,b)\begin{equation} R_e \equiv \frac{\bar{\rho} U L}{\mu}, \quad R_m \equiv \frac{U L}{\eta} , \end{equation}

where $L$ is a characteristic length, $\bar {\rho }$ is a typical density and $U$ a characteristic speed of the system.

We may now inquire: How does the value of those numbers affect the conservation of non-barotropic cross-helicity? To do this, we write each physical variable $g$ as a multiplication of a characteristic value $\bar {g}$ and a dimensionless variable $g'$ in the form

(3.32a-c)\begin{equation} g = \bar{g} g' \quad \Rightarrow \quad \boldsymbol{x} = L \boldsymbol{x}', \quad \boldsymbol{v} = U \boldsymbol{v}', \quad t = \bar{t} t'. \end{equation}

The above equation suggests the following choice of $\bar {t}$:

(3.33)\begin{equation} \bar{t} \equiv \frac{L}{U}. \end{equation}

Similarly, we write

(3.34a-e)\begin{equation} \rho = \bar{\rho} \rho', \quad T = \bar{T} T', \quad \sigma = \bar{\sigma}\sigma', \quad \boldsymbol{B} = \bar{B}\boldsymbol{B}', \quad \boldsymbol{J} = \bar{J} \boldsymbol{J}'. \end{equation}

Equation (3.2) suggests the following definition of $\bar {\sigma }$:

(3.35)\begin{equation} \bar{\sigma} \equiv \bar{T}\,\bar{t} = \frac{\bar{T} L}{U}, \end{equation}

and (2.7) suggests the following definition of $\bar {J}$:

(3.36)\begin{equation} \bar{J}\equiv \frac{\bar{B}}{L}. \end{equation}

For the viscosity tensor, we use a double prime notation (as we already used a single prime notation to distinguish the viscosity tensor from the $\sigma$ scalar). Thus

(3.37a,b)\begin{equation} \sigma_{ik}^{\prime} = \bar{\sigma}' \sigma_{ik}^{\prime\prime}, \quad \sigma_{ik}^{\prime\prime}\equiv \frac{\partial v'_i}{\partial x'_k}+\frac{\partial v'_k}{\partial x'_i}-\frac{2}{3}\delta_{ik}\frac{\partial v'_l}{\partial x'_l}. \end{equation}

It follows from (2.6) that

(3.38)\begin{equation} \bar{\sigma}' = \frac{\mu U}{L} = \frac{\bar{\rho} U^2}{R_e}. \end{equation}

We will also need the magnetic and kinetic energy expressions in dimensionless form

(3.39)\begin{equation} \left.\begin{gathered} E_k = \frac{1}{2} \int \rho \boldsymbol{v}^2 \,{\rm d}^3 x = \bar{\rho} U^2 L^3 E'_k, \quad E'_k \equiv \frac{1}{2} \int \rho' \boldsymbol{v}'^2 \,{\rm d}^3 x' \\ E_m = \frac{1}{8 {\rm \pi}} \int \boldsymbol{B}^2 \,{\rm d}^3 x = \bar{B}^2 L^3 E'_m, \quad E'_m \equiv \frac{1}{8 {\rm \pi}} \int \boldsymbol{B}'^2 \,{\rm d}^3 x'. \end{gathered}\right\}\end{equation}

Finally, we shall look at the amount of heat $Q_c$ that is conducted into the volume (which is not equal to the total change in heat in the volume as heat may be produced by viscosity and ohmic losses; see (2.5)). We may write the heat flux density as

(3.40a-c)\begin{equation} \boldsymbol{q} = \bar{q} \boldsymbol{q}', \quad \boldsymbol{q}' \equiv \boldsymbol{\nabla'} T', \quad \bar{q}\equiv k \frac{\bar{T}}{L}. \end{equation}

Thus the rate of change of $Q_c$ is (in dimensional and dimensionless form)

(3.41a,b)\begin{equation} \frac{{\rm d} Q_c}{{\rm d} t} ={-}\oint \boldsymbol{q} \boldsymbol{\cdot} {\rm d} \boldsymbol{S} ={-} L^2 \bar{q} \oint \boldsymbol{q}' \boldsymbol{\cdot} {\rm d} \boldsymbol{S}', \quad \frac{{\rm d} Q'_c}{{\rm d} t'} ={-}\oint \boldsymbol{q}' \boldsymbol{\cdot} {\rm d} \boldsymbol{S}' ,\end{equation}

where we integrate over a surface encapsulating the flow. Writing as usual

(3.42)\begin{equation} \bar{Q}_c = \frac{Q_c}{Q'_c} = \bar{q}\frac{L^3}{U} = k \bar{T} \frac{L^2}{U} .\end{equation}

Having defined the above quantities, we may write the non-barotropic cross-helicity as

(3.43a-c)\begin{equation} H_{{\rm CNB}} = \bar{H}_{{\rm CNB}} H'_{{\rm CNB}}, \quad \bar{H}_{{\rm CNB}} \equiv L^3U \bar{B}, \quad H'_{{\rm CNB}} \equiv \int {\rm d}^3x'\,\boldsymbol{v'}_t \boldsymbol{\cdot} \boldsymbol{B'}. \end{equation}

Thus

(3.44)\begin{equation} \frac{{\rm d} H_{{\rm CNB}}}{{\rm d}t} = L^2U^2 \bar{B}\frac{{\rm d}H'_{{\rm CNB}}}{{\rm d}t'} \quad \Rightarrow \quad \frac{{\rm d} H'_{{\rm CNB}}}{{\rm d}t'} = \frac{1}{L^2U^2 \bar{B}} \frac{{\rm d} H_{{\rm CNB}}}{{\rm d}t}. \end{equation}

It follows that (3.30) can be written in the form

(3.45)\begin{align} \frac{{\rm d} H'_{{\rm CNB}}}{{\rm d}t'} & = \frac{1}{L^2U^2 \bar{B}} \frac{{\rm d} H_{{\rm CNB}}}{{\rm d}t} = \int\left\{ \frac{1}{R_m} \left[ \frac{\boldsymbol{v}'_t \boldsymbol{\cdot} \nabla^{'2} \boldsymbol{B}'}{4{\rm \pi}} + \frac{J^{'2}}{\rho' T'} \boldsymbol{B}' \boldsymbol{\cdot} \boldsymbol{\nabla}' \sigma' \frac{E'_k}{E'_m} \frac{E_m}{E_k} \right] \right. \nonumber\\ & \quad + \frac{E'_k}{Q'_c} \frac{Q_c}{E_k} (\boldsymbol{B}' \boldsymbol{\cdot} \boldsymbol{\nabla}' \sigma')\frac{\nabla^{'2T'}}{\rho' T'} \nonumber\\ & \quad + \left. \frac{1}{R_e} \left[\frac{B'_i}{\rho'}\frac{\partial \sigma_{ik}^{\prime\prime}}{\partial x'_k} + (\boldsymbol{B}' \boldsymbol{\cdot} \boldsymbol{\nabla}' \sigma') \frac{\sigma_{ik}^{\prime\prime}}{\rho' T'}\frac{\partial v'_{i}}{\partial x'_{k}}\right] \right\}{\rm d}^3x'. \end{align}

Thus, generally speaking, non-barotropic cross-helicity will change slowly for flows with both high Reynolds and high magnetic Reynolds numbers in which the heat conducted is small with respect to the kinetic energy of the flow. Indeed ‘Increasing cross helicity with fixed fluctuation energy increases the time required for energy to cascade to smaller scales, reduces the cascade power, and increases the anisotropy of the small-scale fluctuations’ (Chandran Reference Chandran2008). This has implications for the solar wind and solar corona (Chandran Reference Chandran2008).

It will also be of interest to study how another topological invariant, i.e. the magnetic helicity, changes for a flow of high Reynolds number. And how the rate of dissipation differs between those two quantities. Brief background regarding magnetic helicity is given below for the sake of the reader's convenience. The magnetic helicity is defined as

(3.46)\begin{equation} H_M \equiv \int \boldsymbol{A} \boldsymbol{\cdot} \boldsymbol{B} \,{\rm d}^3 x, \end{equation}

where $\boldsymbol {A}$ is the magnetic vector potential, defined such that

(3.47)\begin{equation} \boldsymbol{B}=\boldsymbol{\nabla} \times \boldsymbol{A}. \end{equation}

For an illustration of specific helical magnetic fields see Yahalom & Lynden-Bell (Reference Yahalom and Lynden-Bell2008). After taking the temporal derivative of the magnetic helicity and simplifying the expressions, we obtain the well-known relation

(3.48)\begin{equation} \frac{{\rm d}H_M}{{\rm d}t}={-}2\eta\int \boldsymbol{J}\boldsymbol{\cdot}\boldsymbol{B}\,{\rm d}^3 x. \end{equation}

The above relation has been verified by many authors (Biskamp Reference Biskamp1997; Akhmet'ev, Kunakovskaya & Kutvitskii Reference Akhmet'ev, Kunakovskaya and Kutvitskii2009; Priest Reference Priest2014; Verma Reference Verma2019, Reference Verma2021). It is clear from (3.48) that, generally speaking, the magnetic diffusivity leads to the non-conservation of magnetic helicity in non-ideal MHD. On the other hand, neither viscosity nor heat conductivity affect the conservation of magnetic helicity. We can easily recover the ideal MHD condition by putting magnetic diffusivity to zero, in which case it is evident that the magnetic helicity is conserved. Furthermore, the above result also shows that magnetic helicity is conserved even in non-ideal flows if the currents are orthogonal to the magnetic field i.e. $\boldsymbol {J}\boldsymbol {\cdot }\boldsymbol {B}=0$. In this case

(3.49)\begin{equation} \frac{{\rm d}H_M}{{\rm d}t}=0. \end{equation}

If the magnetic field and magnetic current density are not strictly orthogonal then the magnetic helicity is only approximately conserved.

The above result can also be expressed in a dimensionless form in which we write the vector potential as

(3.50a,b)\begin{equation} \boldsymbol{A} = \bar{A} \boldsymbol{A}', \quad \bar{A} = \bar{B} L. \end{equation}

And thus

(3.51a-c)\begin{equation} H_M = \bar{H}_M H'_M, \quad H'_M \equiv \int \boldsymbol{A'} \boldsymbol{\cdot} \boldsymbol{B'} \,{\rm d}^3 x', \quad \bar{H}_M \equiv \bar{B}^2 L^4. \end{equation}

For a similar dimensional analysis of magnetic helicity see Russell et al. (Reference Russell, Yeates, Hornig and Wilmot-Smith2015, (17)). It is straightforward to see that

(3.52)\begin{equation} \frac{{\rm d}H'_M}{{\rm d}t'}={-}\frac{2}{R_m} \int \boldsymbol{J'}\boldsymbol{\cdot}\boldsymbol{B'}\,{\rm d}^3 x'. \end{equation}

It follows that an approximate conservation of magnetic helicity is achievable at high magnetic Reynolds number and is independent ofthe values of the Reynolds number and total conducted heat. Thus the phenomenon of magnetic helicity conservation is much more general.

4.1. The ideal case

We introduce a set of standard cylindrical coordinates $R,\phi,z$, where $\hat {R},\hat {\phi },\hat {z}$ are the corresponding unit vectors. We further assume an MHD flow of uniform density $\rho$ confined between the internal and external radii $a_{\textrm {in}}$ and $a$, respectively, i.e. $a_{\textrm {in}} < R < a$. Furthermore assume that the flow contains a helical stratified stationary magnetic field

(4.1)\begin{equation} \boldsymbol{B} = \left\{\begin{array}{@{}ll} 2 B_{\bot}\left(1-\dfrac{R}{a}\right)\hat{\phi} + B_{z0}\hat{z} & a_{{\rm in}}< R< a \\[10pt] 0 & {\rm otherwise}, \end{array} \right. \end{equation}

in which $B_{z0},B_{\bot }$ are constants. The magnetic field is contained in a cylinder of radius $a$ and is independent of $z$. Furthermore, we assume that the planes $z=0$ and $z=L$ can be identified such that a topological torus is created. In such a scenario, the only field lines that will be closed will satisfy the relation

(4.2)\begin{equation} \frac{n}{m}=\frac{B_{\bot}}{{\rm \pi} R B_{z0}} \left(1-\frac{R}{a}\right)L, \quad n,m \ {\rm integers}, \end{equation}

while lines not satisfying this relation will be surface filling.

In Yahalom & Qin (Reference Yahalom and Qin2021), we derive a stationary velocity field $\boldsymbol {v}$ that satisfies the stationary ideal versions of (2.4) and (2.2)

(4.3)\begin{gather} \boldsymbol{\nabla} \times (\boldsymbol{v} \times \boldsymbol{B}) = 0, \end{gather}

(4.4)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} (\rho \boldsymbol{v} ) = 0. \end{gather}

There, we arrived at the simple expression

(4.5)\begin{equation} \boldsymbol{v} = v_0 \frac{R}{a} \hat \phi, \end{equation}

where $v_0$ is a constant with dimensions of velocity. The ideal stationary version of (2.1) is given by

(4.6)\begin{equation} \rho (\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla})\boldsymbol{v} ={-}\boldsymbol{\nabla} p + \frac{(\boldsymbol{\nabla} \times \boldsymbol{B}) \times \boldsymbol{B}}{4 {\rm \pi}} . \end{equation}

This can be solved by the pressure function

(4.7)\begin{equation} p(R) = \frac{B_{\bot}^2}{\rm \pi} \left(3 \frac{R}{a} - \frac{R^2}{a^2}-\ln \left(\frac{R}{a}\right)-2\right) + \frac{1}{2}\rho v_0^2 \left(\frac{R^2}{a^2}-1\right), \quad p(a)=0. \end{equation}

Of course, $p(a_{\textrm {in}}) \neq 0$, and thus one will need a rigid internal cylinder of radius $a_{\textrm {in}}$ that can support such a pressure. We can now calculate the cross-helicity using (3.1), in which we assume a uniform specific entropy such that $\boldsymbol {v}_t = \boldsymbol {v}$. Inserting (4.5) and (4.1) into (3.1) we arrive at the expression

(4.8)\begin{equation} H_{{\rm CNBI}} = \frac{\rm \pi}{3} v_0 B_{\bot} \frac{L}{a^2} \left[a^4 - a_{{\rm in}}^3(4a -3 a_{{\rm in}})\right]. \end{equation}

In order to calculate the magnetic helicity, we need to calculate a vector potential, one possibility is given by

(4.9)\begin{equation} \boldsymbol{A} = \left\{\begin{array}{ll} \dfrac{1}{2} B_{z0} R \hat{\phi} -2 B_{\bot} R \left(1-\dfrac{R}{2 a}\right) \hat{z} & a_{{\rm in}}< R< a \\[9pt] 0 & {\rm otherwise}. \end{array} \right. \end{equation}

Inserting (4.1) and (4.9) into (3.46) we arrive at the result

(4.10)\begin{equation} H_{MI} ={-} \tfrac{2}{3} {\rm \pi}L B_{z0} B_{\bot} (a-a_{{\rm in}}) \left[a^2 + a_{{\rm in}} a +a_{{\rm in}}^2 \right]. \end{equation}

Obviously, both magnetic helicity and magnetic cross-helicity do not change in time. The situation, however, is quite different when one considers non-ideal processes such as magnetic diffusion.

4.2. The non-ideal case

Let us assume a non-ideal magnetic diffusion; the magnetic field $\boldsymbol {B}_T$ will obviously be different from $\boldsymbol {B}$ and we may write it in the following form:

(4.11)\begin{equation} \boldsymbol{B}_T = \boldsymbol{B} + \eta \boldsymbol{B}_1. \end{equation}

In the above, $\boldsymbol {B}$ is given in (4.1) and is thus a stationary solution of an ideal MHD configuration. If we take the typical scale to be $a$ and the typical velocity to be $v_0$, we can write the magnetic Reynolds number in the form

(4.12)\begin{equation} R_m = \frac{v_0 a}{\eta}, \quad \Rightarrow \quad \eta = \frac{v_0 a}{R_m} . \end{equation}

Thus, if we take the typical size of the magnetic fields $\boldsymbol {B}_T$ and $\boldsymbol {B}$ to be $\bar {B}_T = \bar {B} = B_{z0}$ and the typical size of $\boldsymbol {B}_1$ to be $\bar {B}_1 = {B_{z0}}/{v_0 a}$, we may write

(4.13)\begin{equation} \boldsymbol{B}'_T = \boldsymbol{B}' + \frac{1}{R_m} \boldsymbol{B}'_1 . \end{equation}

We shall assume that the magnetic Reynolds number is large, such that the non-ideal correction is small. Now, $\boldsymbol {B}_T$ must satisfy (2.4)

(4.14)\begin{equation} \frac{\partial \boldsymbol{B_T}}{\partial t}=\boldsymbol{\nabla} \times (\boldsymbol{v}\times \boldsymbol{B_T}) +\frac{\eta}{4{\rm \pi}}\nabla^2\boldsymbol{B_T} ,\end{equation}

assuming that the velocity field is given by (4.5), and taking int account that $\boldsymbol {B}$ is stationary, we arrive at

(4.15)\begin{equation} \eta \frac {\partial \boldsymbol{B_1}}{\partial t}=\eta \boldsymbol{\nabla} \times (\boldsymbol{v}\times \boldsymbol{B_1}) +\frac{\eta}{4{\rm \pi}}\nabla^2 (\boldsymbol{B} + \eta \boldsymbol{B_1}). \end{equation}

Thus, $\eta$ can be cancelled out and we obtain

(4.16)\begin{equation} \frac{\partial \boldsymbol{B_1}}{\partial t}= \boldsymbol{\nabla} \times (\boldsymbol{v}\times \boldsymbol{B_1}) +\frac{1}{4{\rm \pi}}\nabla^2 (\boldsymbol{B} + \eta \boldsymbol{B_1}) . \end{equation}

Therefore, the term $\eta \boldsymbol {B_1} = ({B_{z0}}/{R_m})\boldsymbol {B}'_1$ can be neglected for high magnetic Reynolds numbers. Thus, we arrive at the equation

(4.17)\begin{equation} \frac{\partial\boldsymbol{B_1}}{\partial t}= \boldsymbol{\nabla} \times (\boldsymbol{v}\times \boldsymbol{B_1}) +\frac{1}{4{\rm \pi}}\nabla^2 \boldsymbol{B} .\end{equation}

The source term of the above equation is, according to (4.1),

(4.18)\begin{equation} \frac{1}{4{\rm \pi}}\nabla^2 \boldsymbol{B} ={-}\frac{B_{\bot}}{2 {\rm \pi}R^2} \hat{\phi},\end{equation}

for every point $R< a$. Moreover, it is easy to show that, if at a specified time $t=0$ we have $\boldsymbol {B_1} (\boldsymbol {x}, 0) = 0$, it follows that $B_{1R} (\boldsymbol {x}, t) = B_{1z} (\boldsymbol {x}, t) = 0$ for any time $t$. The equation for $B_{1\phi }$ is thus

(4.19)\begin{equation} \frac {\partial B_{1\phi}}{\partial t}={-}\frac{B_{\bot}}{2 {\rm \pi}R^2}, \end{equation}

which can be trivially integrated

(4.20)\begin{equation} B_{1\phi} ={-}\frac{B_{\bot}}{2 {\rm \pi}R^2} t . \end{equation}

Thus the total magnetic field is

(4.21)\begin{equation} \boldsymbol{B}_T = \left\{\begin{array}{@{}ll} 2 B_{\bot}\left(1-\dfrac{R}{a}-\dfrac{\eta}{4 {\rm \pi}R^2} t\right)\hat{\phi} + B_{z0}\hat{z} & a_{{\rm in}}< R< a \\[10pt] 0 & {\rm otherwise}. \end{array} \right.\end{equation}

And thus the current density can be calculated using (2.7) to be

(4.22)\begin{equation} \boldsymbol{J}_T = \left\{\begin{array}{@{}ll} \dfrac{B_{\bot}}{2 {\rm \pi}R} \left( 1-\dfrac{2R}{a}+ \dfrac{\eta}{4 {\rm \pi}R^2} t \right) \hat z & a_{{\rm in}}< R< a \\[10pt] 0 & {\rm otherwise}. \end{array}\right. \end{equation}

Thus one can calculate the time-dependent pressure using (2.1)

(4.23)\begin{align} p(R,t) & = \frac{B_{\bot}^2}{\rm \pi} \left(3 \frac{R}{a} - \frac{R^2}{a^2}-\ln \left(\frac{R}{a}\right)-2 + \frac{\eta t}{4 {\rm \pi}a} \left(\frac{1}{R} - \frac{1}{a}\right) - \frac{\eta^2 t^2}{96 {\rm \pi}^2} \left(\frac{1}{R^6} - \frac{1}{a^6}\right)\right) \nonumber\\ & \quad + \frac{1}{2}\rho v_0^2 \left(\frac{R^2}{a^2}-1\right), \quad p(a,t)=0. \end{align}

Thus the internal cylinder must sustain the above time-dependent pressure.

Now, to calculate the magnetic helicity we need a vector potential; this can be similarly obtained as in the ideal case in the form

(4.24)\begin{equation} \boldsymbol{A}_T = \left\{\begin{array}{@{}ll} \dfrac{1}{2} B_{z0} R \hat{\phi} -2 B_{\bot} R \left(1-\dfrac{R}{2 a}+\dfrac{\eta t}{4 {\rm \pi}R^2}\right)\hat{z} & a_{{\rm in}}< R< a \\[10pt] 0 & {\rm otherwise}. \end{array} \right.\end{equation}

Inserting (4.21) and (4.24) into (3.46) will result in a time-dependent magnetic helicity

(4.25)\begin{equation} H_{{\rm MT}} ={-} \frac{2}{3} {\rm \pi}L B_{z0} B_{\bot} (a-a_{{\rm in}}) \left[a^2 + a_{{\rm in}} a +a_{{\rm in}}^2 +\frac{9 \eta t}{4 {\rm \pi}} \right]. \end{equation}

The time derivative of the above magnetic helicity is

(4.26)\begin{equation} \frac{{\rm d} H_{MT}}{{\rm d}t} ={-} \frac{3}{2} L B_{z0} B_{\bot} (a-a_{{\rm in}}) \eta. \end{equation}

Similarly, inserting (4.21) and (4.5) into (3.1) will result in a time-dependent cross-helicity

(4.27)\begin{equation} H_{{\rm CNBT}} = 4 {\rm \pi}v_0 B_{\bot} \frac{L}{a} \left[\frac{1}{3}(a^3 -a_{{\rm in}}^3)- \frac{1}{4a} (a^4 -a_{{\rm in}}^4) - \frac{\eta t}{4 {\rm \pi}} (a -a_{{\rm in}})\right] ,\end{equation}

with a time derivative of

(4.28)\begin{equation} \frac{{\rm d}H_{{\rm CNBT}}}{{\rm d}t} ={-} v_0 B_{\bot} \frac{L}{a} (a-a_{{\rm in}}) \eta. \end{equation}

It is interesting to compare which of the topological quantities is better preserved given some high Reynolds number (and neglecting viscosity and heat conduction). To this end we compare

(4.29a-c)\begin{equation} \left|\frac{1}{H_{{\rm MI}}} \frac{{\rm d} H_{MT}}{{\rm d}t'}\right|=\frac{9}{4 {\rm \pi}R_m (1+a'_{{\rm in}}+{a'}_{{\rm in}}^2)}, \quad a'_{{\rm in}} \equiv \frac{a_{{\rm in}}}{a},\quad t' = t \frac{v_0}{a}, \end{equation}

with

(4.30)\begin{equation} \left|\frac{1}{H_{{\rm CNBI}}} \frac{{\rm d} H_{{\rm CNBT}}}{{\rm d}t'}\right|= \frac{3 (1-a'_{{\rm in}})}{{\rm \pi} R_m \left[1 - {a'}_{{\rm in}}^3(4-3{a'}_{{\rm in}})\right]}. \end{equation}

Both the above quantities are inversely proportional to the magnetic Reynolds number $R_m$. The results for $Rm=1$ are depicted in figure 1, which illustrates that the relative rates of change of both quantities are about the same, with a slight advantage to the magnetic helicity depending on the particular geometry of the configuration. It is shown that, for the right pressure profile, both magnetic helicity and cross-helicity are of comparable importance. The higher the magnetic Reynolds number, the more ideal the flow is.

Figure 1.

Comparison of $|({1}/{H_{\textrm {MI}}} )({\textrm {d} H_{\textrm {MT}}}/{\textrm {d}t'})|$ with $|({1}/{H_{\textrm {CNBI}}})( {\textrm {d}H_{\textrm {CNBT}}}/{\textrm {d}t'})|$ for $R_m=1$.