The influence of temperature dynamics and dynamic finite ion Larmor radius effects on seeded high amplitude plasma blobs

The polarisation equation is the gyrofluid expression of quasi neutrality and reads

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{n}_{\text{e}}}- \Gamma _{1,\text{i}}^{\dagger}{{N}_{\text{i}}}=\boldsymbol{}\nabla \centerdot \left(\frac{{{N}_{\text{i}}}}{{{ \Omega }_{\text{i}}}B}\boldsymbol{}{{\nabla}_{\bot}}\phi \right), \end{array}\end{eqnarray} \tag{ 1 }$$

where n_e is the electron density, N_i is the ion gyro-centre density, ϕ is the electrostatic potential and ${{ \Omega }_{\text{i}}}=eB/{{m}_{\text{i}}}$ is the ion gyro-frequency. We note here that as a consequence of the gyro-centre transformation [39] the ion gyro-centre density N_i is not to be confused with the ion particle density ${{n}_{\text{i}}}\ne {{N}_{\text{i}}}$. The perpendicular gradient is defined by $\boldsymbol{}{{\nabla}_{\bot}}\equiv -\widehat{\boldsymbol{b}}\times \left(\widehat{\boldsymbol{b}}\times \boldsymbol{}\nabla \right)$ and the unit vector in the magnetic field direction by $\widehat{\boldsymbol{b}}\equiv \boldsymbol{B}/B$. The mass ratio between the electrons and ions is small, permitting us to neglect electron FLR and finite electron inertia effects. This casts the electron gyro-centre variables into common fluid variables $\left({{N}_{\text{e}}},{{T}_{\text{e}}}\right)\equiv \left({{n}_{\text{e}}},{{t}_{\text{e}}}\right)$. The sum of the electron density n_e and the FLR corrected ion gyro-centre density N_i is referred to as the nonlinear polarisation charge density (right hand side of equation (1)). In the polarisation equation (1) no thin-layer approximation $\boldsymbol{}\nabla \centerdot \left(\frac{{{N}_{\text{i}}}}{{{ \Omega }_{\text{i}}}B}\boldsymbol{}{{\nabla}_{\bot}}\phi \right)\approx \frac{{{N}_{\text{i}0}}}{{{ \Omega }_{\text{i}0}}{{B}_{0}}}\boldsymbol{}\nabla _{\bot}^{2}\phi$ is applied and the full nonlinear polarisation charge density is retained. Here, and in all other abbreviations the subscript '0' denotes a constant background quantity. We note that FLR corrections to the polarisation density are included in delta-f gyrokinetic and gyrofluid models [27], but are omitted in full-F gyrokinetic as well as full-F gyrofluid models. Therefore, even if we invoked the thin-layer approximation to the full-F model, it would only agree with delta-f gyrofluid models in the LWL [28]. In gyrofluid models the ion polarisation drift enters via the nonlinear polarisation charge density. The gyroaveraging operators appear as the Padé approximants [41]

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{ \Gamma }_{1,\text{i}}}=\frac{1}{1-\frac{\rho _{\text{i}}^{2}}{2}\boldsymbol{}\nabla _{\bot}^{2}}, \end{array}\end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \Gamma _{1,\text{i}}^{\dagger}=\frac{1}{1-\boldsymbol{}\nabla _{\bot}^{2}\frac{\rho _{\text{i}}^{2}}{2}}. \end{array}\end{eqnarray} \tag{ 2b }$$

Note here the temperature and magnetic field dependence of the ion gyro-radius ${{\rho}_{\text{i}}}=\sqrt{{{T}_{\text{i}}}/{{m}_{\text{i}}}}/{{ \Omega }_{\text{i}}}$, which introduces dynamic FLR effects ${{\rho}_{\text{i}}}\left({{T}_{\text{i}}},B\right)$. In contrast to isothermal gyrofluid models with constant FLR effects ${{\rho}_{\text{i}}}\left({{T}_{\text{i}0}},{{B}_{0}}\right)$ [28] the ${{ \Gamma }_{1,\text{i}}}$ operator is no longer self-adjoint ${{ \Gamma }_{1,\text{i}}}\ne \Gamma _{1,\text{i}}^{\dagger}$ and the ${{ \Gamma }_{2,\text{i}}}\equiv \frac{{{\rho}_{\text{i}}}}{2}\frac{\partial {{ \Gamma }_{1,\text{i}}}}{\partial {{\rho}_{\text{i}}}}$ operator appears in the gyrofluid moment equations due to spatial variations in the temperature or magnetic field

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{ \Gamma }_{2,\text{i}}}=\frac{\frac{\rho _{\text{i}}^{2}}{2}\boldsymbol{}\nabla _{\bot}^{2}}{{{\left(1-\frac{\rho _{\text{i}}^{2}}{2}\boldsymbol{}\nabla _{\bot}^{2}\right)}^{2}}}, \end{array}\end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \Gamma _{2,\text{i}}^{\dagger}=\frac{\boldsymbol{}\nabla _{\bot}^{2}\frac{\rho _{\text{i}}^{2}}{2}}{{{\left(1-\boldsymbol{}\nabla _{\bot}^{2}\frac{\rho _{\text{i}}^{2}}{2}\right)}^{2}}}. \end{array}\end{eqnarray} \tag{ 3b }$$

The FLR corrected ion gyro-centre density $\Gamma _{1}^{\dagger}{{N}_{\text{i}}}$ in the polarisation equation (1) is the averaged charge contribution at a position $\boldsymbol{r}$ from gyro-centres whose gyro-orbits intersect $\boldsymbol{r}$. In case of constant FLR effects the mean gyro-orbit is constant and these gyro-centres are placed on a circle with constant radius ${{\rho}_{\text{i}}}\left({{T}_{\text{i}0}},{{B}_{0}}\right)$ at position $\boldsymbol{r}$. On the contrary, for dynamic FLR effects the mean gyro-orbit is spatially varying, so that these gyro-centres are placed on an arbitrary curve. For the simple case of a Gaussian ion gyro-centre pressure blob this curve reduces to a circle with radius ${{\rho}_{\text{i}}}\left({{T}_{\text{i}}},B\right)$ or a deformed circle at position $\boldsymbol{r}$, which is shown in figure 1. As a consequence the curves sample different ion gyro-centre densities for constant or dynamic FLR effects. This, in comparison to constant FLR effects, effectively increases or decreases the polarisation charge density in the blob centre or edge respectively.

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Due to the neglect of electron FLR effects the gyroaveraging operators for electrons reduce to ${{ \Gamma }_{1,\text{e}}}= \Gamma _{1,\text{e}}^{\dagger}=1$ and ${{ \Gamma }_{2,\text{e}}}= \Gamma _{2,\text{e}}^{\dagger}=0$.

The time evolution of a gyro-centre density N is governed by the zeroth gyrofluid moment over the gyrokinetic Vlasov equation

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \frac{\partial}{\partial t}N+\boldsymbol{}\nabla \centerdot \left(N\left[\boldsymbol{U}_{{E}}+\boldsymbol{U}_{{\eta}}+\boldsymbol{U}_{{\boldsymbol{}\nabla B}}\right]\right)={{ \Lambda }_{N}}. \end{array}\end{eqnarray} \tag{ 4 }$$

Here, the gyrofluid drifts (denoted with capital U) account for the $\boldsymbol{E}\times \boldsymbol{B}$ drive by the $\boldsymbol{E}\times \boldsymbol{B}$ drift $\boldsymbol{U}_{{E}}$, the interchange drive by the $\boldsymbol{}\nabla B$ drift $\boldsymbol{U}_{{\boldsymbol{}\nabla B}}$ and FLR corrections to the gyro-averaged electric potential due to variations in the temperature or magnetic field by $\boldsymbol{U}_{{\eta}}$:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \,\boldsymbol{U}_{{E}}\equiv \frac{\widehat{\boldsymbol{b}}\times \boldsymbol{}\nabla \psi}{B}, \end{array}\\ &&\ \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \boldsymbol{U}_{{\boldsymbol{}\nabla B}}\equiv \frac{T\widehat{\boldsymbol{b}}\times \boldsymbol{}\nabla \ln B}{qB}, \end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \boldsymbol{U}_{{\eta}}\equiv \frac{\left({{ \Gamma }_{2}}\phi \right)\widehat{\boldsymbol{b}}\times \boldsymbol{}\nabla \eta}{B}. \end{array}\end{eqnarray} \tag{ 7 }$$

Note that as a consequence of the choice of our slab geometry, which is introduced later in this section, no curvature drift appears in the gyrofluid moment equations. In equations (5) and (7) we introduced the gradient of temperature and magnetic field $\boldsymbol{}\nabla \eta$, the generalised potential $\psi$ and the $\boldsymbol{E}\times \boldsymbol{B}$ drift velocity $\boldsymbol{u}_{{E}}$:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \boldsymbol{}\nabla \eta =\boldsymbol{}\nabla \ln B-\boldsymbol{}\nabla \ln T, \end{array}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \psi ={{ \Gamma }_{1}}\phi -\frac{mu_{E}^{2}}{2q}, \end{array}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \boldsymbol{u}_{{E}}=\frac{1}{B}\widehat{\boldsymbol{b}}\times \boldsymbol{}\nabla \phi. \end{array}\end{eqnarray} \tag{ 10 }$$

Here, q is the particle charge. Since the gyrokinetic Lagrangian is taken in the LWL, polarisation effects are retained in the generalised potential ψ of equation (9) only via the $\boldsymbol{E}\times \boldsymbol{B}$ energy term [38]. The dissipative term in the gyrofluid moment equation (4) is a hyperdiffusive term of second order ${{ \Lambda }_{N}}\equiv -\nu \nabla _{\bot}^{4}N$. The second gyrofluid moment yields the partial differential equation for a perpendicular gyro-centre pressure P. The equation is rewritten with the help of the zeroth gyrofluid moment equation (4) into an evolution equation for a perpendicular gyro-centre temperature T:

$$\begin{eqnarray}\begin{array}{*{35}{l}} \frac{\partial}{\partial t}T & +\,T\boldsymbol{}\nabla \centerdot \left(\boldsymbol{U}_{{E,2}}+\boldsymbol{U}_{{\boldsymbol{}\nabla B}}+\boldsymbol{U}_{{\eta}}\right) \\ {} & +\,T\left(\boldsymbol{U}_{{E,2}}+\boldsymbol{U}_{{\boldsymbol{}\nabla B}}+\boldsymbol{U}_{{\eta}}\right)\centerdot \boldsymbol{}\nabla \ln N \\ {} & -\,T\left(\boldsymbol{U}_{{E}}+\boldsymbol{U}_{{E,2}}+2\boldsymbol{U}_{{\boldsymbol{}\nabla B}}\right)\centerdot \boldsymbol{}\nabla \eta ={{ \Lambda }_{T}}. \end{array}\end{eqnarray} \tag{ 11 }$$

Here, the gyrofluid drift $\boldsymbol{U}_{{E,2}}$ captures FLR corrections to the $\boldsymbol{E}\times \boldsymbol{B}$ drift $\boldsymbol{U}_{{E}}$, which emerge due to spatial variations in the gyro-centre temperature T or the magnetic field B:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \boldsymbol{U}_{{E,2}}\equiv \frac{\widehat{\boldsymbol{b}}\times \boldsymbol{}\nabla {{ \Gamma }_{2}}\phi}{B}. \end{array}\end{eqnarray} \tag{ 12 }$$

In the second gyrofluid moment equation (11) a hyperdiffusive term of second order ${{ \Lambda }_{T}}\equiv -\nu \nabla _{\bot}^{4}T$ is added, which together with ${{ \Lambda }_{N}}$ ensures numerical stability.

Our gyrofluid model employs right-handed Cartesian coordinates (x, y, z) and a slab magnetic field $\widehat{\boldsymbol{b}}={{\widehat{\boldsymbol{e}}}_{z}}$ with a radially varying magnitude $\frac{1}{B}=\frac{1}{{{B}_{0}}}\left(1+x/R\right)$. Here, R is the radial distance to the outboard mid-plane and B₀ is the reference magnetic field magnitude. In this simple geometry a more accessible representation of the electron and ion gyrofluid moment equations (4) and (11) is obtained by expressing the set of equations with the help of Poisson brackets and curvature operators. The slab geometry approximation reduces the curvature operator to $\mathcal{K}\left(\,f\right)\equiv \nabla \centerdot \left(\frac{1}{B}\widehat{\boldsymbol{b}}\times \boldsymbol{}\nabla f\right)=-\frac{1}{{{B}_{0}}R}\frac{\partial f}{\partial y}$ and the Poisson bracket to ${{\left[\,f,g\right]}_{xy}}=\frac{\partial f}{\partial x}\frac{\partial g}{\partial y}-\frac{\partial g}{\partial x}\frac{\partial f}{\partial y}$. We rearrange the terms in the gyrofluid moment equations (4) and (11) to highlight the highly nonlinear nature of our system:

$$\begin{eqnarray}\begin{array}{*{35}{l}} \frac{\partial}{\partial t}{{n}_{\text{e}}}= & -\frac{1}{B}{{\left[\phi,{{n}_{\text{e}}}\right]}_{xy}}-{{n}_{\text{e}}}\mathcal{K}\left(\phi \right)+\frac{1}{e}\mathcal{K}\left({{t}_{\text{e}}}{{n}_{\text{e}}}\right)+\,{{ \Lambda }_{{{n}_{\text{e}}}}}, \end{array}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \frac{\partial}{\partial t}{{N}_{\text{i}}}= & -\frac{1}{B}{{\left[{{\psi}_{\text{i}}},{{N}_{\text{i}}}\right]}_{xy}}+\frac{1}{B}{{\left[\ln {{T}_{\text{i}}},{{N}_{\text{i}}}{{\chi}_{\text{i}}}\right]}_{xy}} \\ {} & -\,{{N}_{\text{i}}}\mathcal{K}\left({{\psi}_{\text{i}}}+{{\chi}_{\text{i}}}\right)+{{N}_{\text{i}}}{{\chi}_{\text{i}}}\mathcal{K}\left(\ln {{T}_{\text{i}}}-\ln {{N}_{\text{i}}}\right) \\ {} & -\,\frac{1}{e}\mathcal{K}\left({{T}_{\text{i}}}{{N}_{\text{i}}}\right)+{{ \Lambda }_{{{N}_{\text{i}}}}}, \end{array}\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \frac{\partial}{\partial t}{{t}_{\text{e}}}= & -\frac{1}{B}{{\left[\phi,{{t}_{\text{e}}}\right]}_{xy}}-{{t}_{\text{e}}}\mathcal{K}\left(\phi \right)+\frac{3{{t}_{\text{e}}}}{e}\mathcal{K}\left({{t}_{\text{e}}}\right) \\ {} & +\,\left(\frac{t_{\text{e}}^{2}}{e}\right)\mathcal{K}\left(\ln {{n}_{\text{e}}}\right)+{{ \Lambda }_{{{t}_{\text{e}}}}}, \end{array}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \frac{\partial}{\partial t}{{T}_{\text{i}}}= & -\frac{1}{B}{{\left[{{\psi}_{\text{i}}}+2{{\chi}_{\text{i}}},{{T}_{\text{i}}}\right]}_{xy}} \\ {} & -\,\frac{{{T}_{\text{i}}}{{\chi}_{\text{i}}}}{B}{{\left[\ln {{\chi}_{\text{i}}}-\ln {{T}_{\text{i}}},\ln {{N}_{\text{i}}}\right]}_{xy}} \\ {} & -\,{{T}_{\text{i}}}\mathcal{K}\left({{\psi}_{\text{i}}}+3{{\chi}_{\text{i}}}\right)-\left(\frac{3{{T}_{\text{i}}}}{e}-{{\chi}_{\text{i}}}\right)\mathcal{K}\left({{T}_{\text{i}}}\right) \\ {} & -\,\left(\frac{T_{\text{i}}^{2}}{e}+{{T}_{\text{i}}}{{\chi}_{\text{i}}}\right)\mathcal{K}\left(\ln {{N}_{\text{i}}}\right)+{{ \Lambda }_{{{T}_{\text{i}}}}}. \end{array}\end{eqnarray} \tag{ 16 }$$

Here, we introduced ${{\chi}_{\text{i}}}\equiv {{ \Gamma }_{2,\text{i}}}\phi$. In the isothermal limit the temperatures are constant $\left({{t}_{\text{e}}},{{T}_{\text{i}}}\right)=\left({{t}_{\text{e}0}},{{T}_{\text{i}0}}\right)$ and second order FLR effects vanish so that ${{ \Gamma }_{1,\text{i}}}= \Gamma _{1,\text{i}}^{\dagger}$ and ${{ \Gamma }_{2,\text{i}}}= \Gamma _{2,\text{i}}^{\dagger}=0$. From this we obtain the set of full-F gyrofluid equations, which was studied in [28].

The energy $\mathcal{E}$ of our system can be determined by integrating the sum of the zeroth gyrofluid moment equation multiplied by the factor $\left(T+q\psi \right)$ and the second gyrofluid moment equation over the domain. We obtain the energy theorem $\frac{\partial}{\partial t}\mathcal{E}= \Lambda$ for electrons and ions by neglecting boundary terms:

$$\begin{eqnarray}\begin{array}{*{35}{l}} \mathcal{E}= & {\int}^{}\text{d}\boldsymbol{x}\left({{n}_{\text{e}}}{{t}_{\text{e}}}+{{N}_{\text{i}}}{{T}_{\text{i}}}+\frac{{{m}_{\text{i}}}{{N}_{\text{i}}}u_{E}^{2}}{2}\right),\,\,\, \end{array}\,\,\,\,\,\,\,\,\,\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \Lambda = & {\int}^{}\text{d}\boldsymbol{x}\left[\left({{t}_{\text{e}}}-e\phi \right){{ \Lambda }_{{{n}_{\text{e}}}}}+\left({{T}_{\text{i}}}+e{{\psi}_{\text{i}}}\right){{ \Lambda }_{{{N}_{\text{i}}}}} \right. \\ {} & \left. +\,{{n}_{\text{e}}}{{ \Lambda }_{{{t}_{\text{e}}}}}+\left(1+\frac{e}{{{T}_{\text{i}}}}{{\chi}_{\text{i}}}\right){{N}_{\text{i}}}{{ \Lambda }_{{{T}_{\text{i}}}}}\right]. \end{array}\end{eqnarray} \tag{ 18 }$$

The energy consists of the internal (thermal) energy densities for electrons and ions and the $\boldsymbol{E}\times \boldsymbol{B}$ energy density. The dissipative terms ${{ \Lambda }_{N}},{{ \Lambda }_{T}}$ enter the energy theorem via $\Lambda$. The derived energy theorem resembles those in [38, 42].

In order to understand the relation between gyro-centre and particle fields and to show the correspondence to drift-fluid models explicitly we derive the inherent vorticity density equation of the presented full-F gyrofluid model in the LWL [40]. Thus we first invert the polarisation equation (1) and apply the LWL. This yields an expression for N_i, which depends on the physical meaningful variables n_e, t_i and ϕ [42]:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{N}_{\text{i}}} & \approx {{n}_{\text{e}}}-\boldsymbol{}\nabla _{\bot}^{2}\left(\frac{{{n}_{\text{e}}}{{t}_{\text{i}}}}{2{{m}_{\text{i}}} \Omega _{\text{i}}^{2}}\right)-\boldsymbol{}\nabla \centerdot \left(\frac{{{n}_{\text{e}}}}{B{{ \Omega }_{\text{i}}}}\boldsymbol{}{{\nabla}_{\bot}}\phi \right) \\ {} & \equiv {{n}_{\text{e}}}\left(1-\frac{{{\omega}^{\ast}}}{{{ \Omega }_{\text{i}}}}\right). \end{array}\end{eqnarray} \tag{ 19 }$$

Here, we defined the sum of the $\boldsymbol{E}\times \boldsymbol{B}$ vorticity plus half of the ion diamagnetic vorticity as

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\omega}^{\ast}} & =\frac{{{ \Omega }_{\text{i}}}}{{{n}_{\text{e}}}}\left[\boldsymbol{}\nabla \centerdot \left(\frac{{{n}_{\text{e}}}}{B{{ \Omega }_{\text{i}}}}\boldsymbol{}{{\nabla}_{\bot}}\phi \right)+\boldsymbol{}\nabla _{\bot}^{2}\left(\frac{{{n}_{\text{e}}}{{t}_{\text{i}}}}{2{{m}_{\text{i}}} \Omega _{\text{i}}^{2}}\right)\right] \\ {} & \equiv {{ \Omega }_{E}}+\frac{1}{2}{{ \Omega }_{d}}. \end{array}\end{eqnarray} \tag{ 20 }$$

The generalised vorticity ${{ \Omega }^{\ast}}={{\omega}^{\ast}}+\frac{1}{2}{{ \Omega }_{d}}$ is the sum of the magnetic field aligned component of the $\boldsymbol{E}\times \boldsymbol{B}$ vorticity and the ion diamagnetic vorticity. We note here that the $\boldsymbol{E}\times \boldsymbol{B}$ vorticity consists of the magnetic field aligned component of the common $\boldsymbol{E}\times \boldsymbol{B}$ vorticity ${{ \Omega }_{E0}}\equiv \widehat{\boldsymbol{b}}\centerdot \boldsymbol{}\nabla \times \boldsymbol{u}_{{E}}=\boldsymbol{}\nabla \centerdot \left(\frac{1}{B}\boldsymbol{}{{\nabla}_{\bot}}\phi \right)$ plus a cross term ${{ \Omega }_{EX}}\equiv \frac{1}{B}\boldsymbol{}\nabla \ln \left(\frac{{{n}_{\text{e}}}}{{{ \Omega }_{\text{i}}}}\right)\centerdot \boldsymbol{}{{\nabla}_{\bot}}\phi$. Analogously the magnetic field aligned component of the ion diamagnetic vorticity consists of ${{ \Omega }_{d0}}\equiv \widehat{\boldsymbol{b}}\centerdot \boldsymbol{}\nabla \times \boldsymbol{u}_{{d}}=\boldsymbol{}\nabla \centerdot \left(\frac{1}{{{ \Omega }_{\text{i}}}{{m}_{\text{i}}}{{n}_{\text{e}}}}\boldsymbol{}{{\nabla}_{\bot}}{{p}_{\text{i}\bot}}\right)$ and a cross term ${{ \Omega }_{dX}}\equiv \frac{{{t}_{\text{i}\bot}}}{{{m}_{\text{i}}}{{ \Omega }_{\text{i}}}}\left[4{{\left(\boldsymbol{}{{\nabla}_{\bot}}\ln {{ \Omega }_{\text{i}}}\right)}^{2}}-2\boldsymbol{}\nabla _{\bot}^{2}\ln {{ \Omega }_{\text{i}}}-\boldsymbol{}{{\nabla}_{\bot}}\ln {{p}_{\text{i}\bot}}\right.$ $\left. \centerdot \left(3\boldsymbol{}{{\nabla}_{\bot}}\ln {{ \Omega }_{\text{i}}}+\boldsymbol{}{{\nabla}_{\bot}}\ln {{n}_{\text{e}}}\right)\right]$ The generalised vorticity density is given by $\mathcal{W}\equiv {{n}_{\text{e}}}{{ \Omega }^{\ast}}$. The LWL vorticity density equation is derived by taking the material derivative $\frac{\text{d}f}{\text{d}t}\equiv \frac{\partial f}{\partial t}+\frac{1}{B}{{\left[\phi,f\right]}_{xy}}$ over the generalised vorticity density

$$\begin{eqnarray}\begin{array}{*{35}{l}} \frac{1}{{{ \Omega }_{\text{i}}}}\frac{\text{d}}{\text{d}t}\mathcal{W}\approx & \frac{1}{e}\mathcal{K}\left(\,{{p}_{\text{e}}}+{{p}_{\text{i}}}\right)-\frac{1}{B}{{\left[\boldsymbol{}{{\nabla}_{\bot}}\phi,\boldsymbol{}{{\nabla}_{\bot}}\left(\rho _{\text{i}}^{2}{{n}_{\text{e}}}\right)\right]}_{xy}} \\ {} & +\,\frac{1}{{{ \Omega }_{\text{i}}}}{{ \Lambda }_{\mathcal{W}}}, \end{array}\end{eqnarray} \tag{ 21 }$$

where we have neglected FLR corrections in curvature operators and nonlinear terms that fall into the LWL. Dynamic FLR effects arise from the ion diamagnetic contribution of the generalised vorticity density $\mathcal{W}$ and the second term on the right hand side of the vorticity density equation (21). The dissipation of the generalised vorticity density is represented by ${{ \Lambda }_{\mathcal{W}}}$, which is discussed in detail in appendix. We note here that the Prandtl number $Pr\equiv \frac{{{\nu}_{{{ \Omega }^{\ast}}}}}{{{\nu}_{T}}}$ and Schmidt number $Sc\equiv \frac{{{\nu}_{{{ \Omega }^{\ast}}}}}{{{\nu}_{N}}}$ are fixed to unity. Hence, the mass diffusivity ${{\nu}_{N}}$, the thermal diffusivity ${{\nu}_{T}}$ and the kinematic viscosity ${{\nu}_{{{ \Omega }^{\ast}}}}$ reduce to a single dissipative parameter ν. In the absence of dissipative terms we can rewrite the vorticity density equation (21) to a form, which equals the divergence of the ion polarisation flux. It reads

$$\begin{eqnarray}\begin{array}{*{35}{l}} \boldsymbol{}\nabla \centerdot \left(\frac{{{n}_{\text{e}}}}{B{{ \Omega }_{\text{i}}}}\frac{\text{d}}{\text{d}t}\boldsymbol{}{{\nabla}_{\bot}}{{\phi}^{\ast}}\right) & \approx \frac{1}{e}\mathcal{K}\left(\,{{p}_{\text{e}}}+{{p}_{\text{i}}}\right), \end{array}\end{eqnarray} \tag{ 22 }$$

and allows a comparison to FLR-corrected drift-fluid models [43, 44]. Here, we defined $\boldsymbol{}{{\nabla}_{\bot}}{{\phi}^{\ast}}\equiv \boldsymbol{}{{\nabla}_{\bot}}\phi +\frac{{{t}_{\text{i}}}}{e}\boldsymbol{}{{\nabla}_{\bot}}\ln \left(\frac{{{p}_{\text{i}}}}{{{m}_{\text{i}}} \Omega _{\text{i}}^{2}}\right)$.

Since gyro-centre fields ${{N}_{\text{i}}},{{T}_{\text{i}}}$ match the particle fields ${{n}_{\text{i}}},{{t}_{\text{i}}}$ only to zeroth order in $k_{\bot}^{2}\rho _{\text{i}}^{2}$ and $e\phi /{{t}_{\text{i}}}$, an accurate initialisation procedure is required in order to avoid any residual $\boldsymbol{E}\times \boldsymbol{B}$ vorticity. Hence, our initialisation routine relies on transformations from the gyro-centre fields N_i, T_i, ${{P}_{\text{i}}}={{N}_{\text{i}}}{{T}_{\text{i}}}$ to the particle fields ${{n}_{\text{e}}}={{n}_{\text{i}}}$, t_i, t_e, ${{p}_{\text{i}}}={{n}_{\text{i}}}{{t}_{\text{i}}}$ without the generation of artificial initial $\boldsymbol{E}\times \boldsymbol{B}$ vorticity. For the ion gyro-centre density and perpendicular temperature we mimic an initial blob by a Gaussian of the form

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{N}_{\text{i}}}\left(\boldsymbol{x},0\right)={{n}_{\text{e}0}}\left[1+A\exp \left(-\frac{{{\left(\boldsymbol{x}-\boldsymbol{x}_{{0}}\right)}^{2}}}{2{{\sigma}^{2}}}\right)\right], \end{array}\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{T}_{\text{i}}}\left(\boldsymbol{x},0\right)={{t}_{\text{i}0}}\left[1+A\exp \left(-\frac{{{\left(\boldsymbol{x}-\boldsymbol{x}_{{0}}\right)}^{2}}}{2{{\sigma}^{2}}}\right)\right], \end{array}\end{eqnarray} \tag{ 24 }$$

with initial amplitude A of the gyro-centre fields N_i and T_i. In order to initialise with zero $\boldsymbol{E}\times \boldsymbol{B}$ vorticity we take the polarisation equation (1) and the equation for p_i [42] in the limit $\phi =0$. This yields the following initial particle fields

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{n}_{\text{e}}}= \Gamma _{1,\text{i}}^{\dagger}{{N}_{\text{i}}}, \end{array}\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{p}_{\text{i}}}=\left(\Gamma _{1,\text{i}}^{\dagger}+ \Gamma _{2,\text{i}}^{\dagger}\right){{P}_{\text{i}}}. \end{array}\end{eqnarray} \tag{ 26 }$$

The initial electron pressure is related to ion pressure via ${{p}_{\text{i}}}={{\tau}_{\text{i}}}{{p}_{\text{e}}}$ so that the initial electron temperature is given by ${{t}_{\text{e}}}={{t}_{\text{i}}}/{{\tau}_{\text{i}}}$. We note here that the initial amplitudes and cross-field sizes of the particle fields n_e, t_e and t_i may differ from the initial parameters A and σ of the gyro-centre fields N_i and T_i due to the manifestation of FLR effects. Hence, the true initial amplitudes $\Delta {{n}_{\text{e}}}\equiv \max \left\{{{n}_{\text{e}}}\left(\boldsymbol{x},0\right)-{{n}_{\text{e}0}}|\boldsymbol{x}\in V\right\}$, $\Delta {{t}_{\text{e}}}\equiv \max \left\{{{t}_{\text{e}}}\left(\boldsymbol{x},0\right)-{{t}_{\text{e}0}}|\boldsymbol{x}\in V\right\}$ and $\Delta {{t}_{\text{i}}}\equiv \max \left\{{{\tau}_{\text{i}}}\left({{t}_{\text{e}}}\left(\boldsymbol{x},0\right)-{{t}_{\text{e}0}}\right)\right.$ $\left. |\boldsymbol{x}\in V\right\}$ are taken into account for the derivation of quantities like the global interchange rate ${{\gamma}_{g}}$, which is defined in the next section 2.6. For the size of the blob we take the initial parameter σ of the gyro-centre fields.

Now, we use the derived vorticity density equation (21) to obtain an estimate for the maximal perpendicular blob velocity ${{V}_{\bot}}$. For this sake we first employ the thin-layer approximation on equation (21) and neglect all terms, which are related to FLR effects or dissipation. Our scaling analysis relies on the ideal interchange rate γ and the blob cross-field size σ as typical time and spatial scale, which results in the typical velocity scale ${{V}_{\bot}}=\gamma \sigma$. The characteristic scale of the generalised vorticity and perpendicular pressure is the ideal interchange rate γ and the total scalar pressure perturbation $\Delta {{p}_{\text{e}}}+ \Delta {{p}_{\text{i}}}$. Following [12] we obtain the explicit expression for the ideal interchange rate $\gamma =\sqrt{\frac{ \Delta {{p}_{\text{e}}}+ \Delta {{p}_{\text{i}}}}{2R{{n}_{\text{e}0}}\sigma {{m}_{\text{i}}}}}$. We note that an initial electron and ion temperature perturbation $\left(\Delta {{t}_{\text{e}}}, \Delta {{t}_{\text{i}}}\right)$ enters the initial electron and ion pressure perturbation according to $\left(\Delta {{p}_{\text{e}}}, \Delta {{p}_{\text{i}}}\right)\sim \left({{n}_{\text{e}0}} \Delta {{t}_{\text{e}}}+{{t}_{\text{e}0}} \Delta {{n}_{\text{e}}}+ \Delta {{n}_{\text{e}}} \Delta {{t}_{\text{e}}},{{n}_{\text{e}0}} \Delta {{t}_{\text{i}}}+{{t}_{\text{i}0}} \Delta {{n}_{\text{e}}}+\right.$ $\left. \Delta {{n}_{\text{e}}} \Delta {{t}_{\text{i}}}\right)$, which in the isothermal limit reduces to $\left(\Delta {{p}_{\text{e}}}, \Delta {{p}_{\text{i}}}\right)\sim \left({{t}_{\text{e}0}} \Delta {{n}_{\text{e}}},{{t}_{\text{i}0}} \Delta {{n}_{\text{e}}}\right)$.

Now, we postulate the factor $\sqrt{1+ \Delta {{n}_{\text{e}}}/{{n}_{\text{e}0}}}$ 'ex post' as a nonlinear correction factor to the blob size σ and the ideal interchange rate γ. As a result the typical spatial and time scale is the effective blob size ${{\sigma}_{g}}\equiv \sigma \sqrt{1+ \Delta {{n}_{\text{e}}}/{{n}_{\text{e}0}}}$ and the global interchange rate [28]

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{\gamma}_{g}}=\sqrt{\frac{ \Delta {{p}_{\text{e}}}+ \Delta {{p}_{\text{i}}}}{2R\left({{n}_{\text{e}0}}+ \Delta {{n}_{\text{e}}}\right)\sigma {{m}_{\text{i}}}}}. \end{array}\end{eqnarray} \tag{ 27 }$$

This form of scaling fits our simulation results best. With the help of the global interchange rate ${{\gamma}_{g}}={{V}_{\bot}}/{{\sigma}_{g}}$ we obtain the scaling law for the maximal perpendicular blob velocity

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{V}_{\bot}}=\sqrt{\frac{\sigma \left(\Delta {{p}_{\text{e}}}+ \Delta {{p}_{\text{i}}}\right)}{2R{{n}_{\text{e}0}}{{m}_{\text{i}}}}}. \end{array}\end{eqnarray} \tag{ 28 }$$

The derived velocity scaling estimate of equation (28) differs for high electron density amplitudes $\Delta {{n}_{\text{e}}}$ from the so called global scaling ${{V}_{\bot,g}}\equiv {{V}_{\bot}}/\sqrt{1+ \Delta {{n}_{\text{e}}}/{{n}_{\text{e}0}}}$ as reported in [28, 45]. However, in [22, 23] the amplitude scaling of equation (28) was verified numerically for high fluctuation amplitudes in the cold ion temperature and isothermal limit. In the isothermal limit our derived velocity scaling law agrees up to a factor $\sqrt{2}$ with the inertial velocity scaling law of [12]. This is reasoned in the low beta $\beta \equiv 2p{{\mu}_{0}}/{{B}^{2}}\ll 1$ approximation of the curvature operator. In this case the magnetic curvature is $\boldsymbol{\kappa }\equiv \widehat{\boldsymbol{b}}\centerdot \boldsymbol{}\nabla \widehat{\boldsymbol{b}}\approx \boldsymbol{}{{\nabla}_{\bot}}\ln B$, whereas for a slab magnetic field the magnetic curvature is $\boldsymbol{\kappa }=0$. Note that equation (28) also coincides with [17] up to a factor 2.

Now, we define a parameter, which is a measure of the FLR strength in the inertial regime. In order to do so we take the ratio of the magnetic field aligned component of the ion diamagnetic vorticity ${{ \Omega }_{d0}}$ to the estimate for the magnetic field aligned component of the $\boldsymbol{E}\times \boldsymbol{B}$ vorticity ${{ \Omega }_{E0}}\sim {{V}_{\bot}}/\sigma$ in the inertial regime. This defines the FLR strength parameter as $\Theta \equiv \frac{{{ \Omega }_{d0}}}{{{ \Omega }_{E0}}}$ oder

$$\begin{eqnarray}\begin{array}{*{35}{l}} \Theta & \approx \sqrt{\frac{ \Delta p_{\text{i}}^{2}}{\left(\Delta {{p}_{\text{e}}}+ \Delta {{p}_{\text{i}}}\right){{t}_{\text{e}0}}{{n}_{\text{e}0}}}\frac{2R\rho _{s0}^{2}}{{{\sigma}^{3}}}}. \end{array}\end{eqnarray} \tag{ 29 }$$

We note here that in the FLR strength parameter $\Theta$ the ion diamagnetic vorticity ${{ \Omega }_{d0}}$ stems from the LWL over the gyroaveraging operator $\Gamma _{1,\text{i}}^{\dagger}$.

In the cold ion limit a finite electron pressure gradient induces an $\boldsymbol{E}\times \boldsymbol{B}$-vorticity dipole due to the $\boldsymbol{}\nabla B$ drift. This accelerates the blob into the radial direction [18]. A steep and vertically extended pressure front develops finally into an up–down symmetric mushroom shape. This symmetry is broken in the late purely nonlinear phase due to the up–down asymmetric nature of the gyrofluid model. We depict this behaviour for electron density n_e, electron temperature t_e and the $\boldsymbol{E}\times \boldsymbol{B}$ vorticity ${{ \Omega }_{E0}}\approx \frac{1}{{{B}_{0}}}\nabla _{\bot}^{2}\phi$ of an exemplary cold blob in figure 2.

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Finite ion temperature effects change the behaviour of the blob fundamentally [27, 28]. We can see this by comparing the blob structures and positions of figures 2 and 3. FLR corrections arise, which contribute to the polarisation charge density in equation (1) and advect electrons and ions with different $\boldsymbol{E}\times \boldsymbol{B}$ drifts $\boldsymbol{u}_{{E}}$ and $\boldsymbol{U}_{{E}}$ (see equations (10) and (5)). Moreover FLR effects manifest themselves by additional drifts (see equations (12) and (7)) in the gyrofluid moment equations and by the ion diamagnetic vorticity ${{ \Omega }_{d}}$ in equation (20). In the initial phase a tilted $\boldsymbol{E}\times \boldsymbol{B}$-vorticity dipole is generated by a finite ion diamagnetic vorticity ${{ \Omega }_{d}}$ and the term $-\frac{1}{B}{{\left[\boldsymbol{}{{\nabla}_{\bot}}\phi,\boldsymbol{}{{\nabla}_{\bot}}\left(\rho _{\text{i}}^{2}{{n}_{\text{e}}}\right)\right]}_{xy}}$ in the generalised vorticity density equation (21) (see figure 4 and [27]). Hence, the motion of the blobs is no more purely radial and an additional poloidal motion in the $\widehat{\boldsymbol{b}}\times \boldsymbol{}\nabla B$ direction is observed. Subsequently, the tilted $\boldsymbol{E}\times \boldsymbol{B}$-vorticity dipole rolls itself up and FLR effects lead to the aligning of $\boldsymbol{E}\times \boldsymbol{B}$ vorticity with ion pressure p_i. The $\boldsymbol{E}\times \boldsymbol{B}$ shear flow produces a differential rotation, which is strongest at the leading edge of the blob. This effect helps to retain the initial rotational symmetry of the blob and decreases the separation of small eddies. As a consequence the blob tends to retain its initial Gaussian structure. This is shown for electron density n_e, ion temperature t_i and the $\boldsymbol{E}\times \boldsymbol{B}$ vorticity of an exemplary hot blob in figure 3. The electron temperature fields match to a great extent the electron density fields.

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In this section we investigate the propagation of isothermal and thermal blobs for various ion temperatures, cross-field sizes and initial amplitudes. The position of the blob is tracked either by its centre of mass (COM) or by its maximal electron density amplitude. We define the COM of a blob by

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \boldsymbol{R}_{{c}}\equiv \frac{1}{M}{\int}^{}\text{d}\boldsymbol{x}\left({{n}_{\text{e}}}-{{n}_{\text{e}0}}\right)\boldsymbol{x}, \end{array}\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} M\equiv {\int}^{}\text{d}\boldsymbol{x}\left({{n}_{\text{e}}}-{{n}_{\text{e}0}}\right). \end{array}\,\,\,\,\,\,\,\,\,\,\end{eqnarray} \tag{ 31 }$$

The COM displacement $\Delta \boldsymbol{R}_{{c}}(t)\equiv \boldsymbol{R}_{{c}}(t)-\boldsymbol{R}_{{c}}(0)=\left(\Delta {{X}_{c}}(t){{,}^{{}}}\right.$ $\left. \Delta {{Y}_{c}}(t)\right){{}^{T}}$ is associated with the time integrated perpendicular particle transport $\boldsymbol{T}_{{{{n}_{\text{e}}}}}(t)\equiv {\int}_{0}^{t}d{{t}^{\prime}}\boldsymbol{}{{\mathbf{\Gamma}}_{{{n}_{\text{e}}}}}\left({{t}^{\prime}}\right)$ via

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \boldsymbol{T}_{{{{n}_{\text{e}}}}}(t)\approx M \Delta \boldsymbol{R}_{{c}}(t). \end{array}\end{eqnarray} \tag{ 32 }$$

Here, we introduced the perpendicular particle transport $\boldsymbol{}{{\mathbf{\Gamma}}_{{{n}_{\text{e}}}}}(t)\equiv {\int}^{}\text{d}\boldsymbol{x}{{n}_{\text{e}}}\left(\boldsymbol{u}_{{E}}+\boldsymbol{u}_{{\boldsymbol{}\nabla B}}\right)$. The position of the maximal electron density amplitude $\boldsymbol{R}_{{\text{max}}}$ is defined implicitly through ${{n}_{\text{e}}}\left(\boldsymbol{R}_{{\text{max}}}\right)\geqslant {{n}_{\text{e}}}\left(\boldsymbol{x}\right)\forall \boldsymbol{x}$ and differs from the COM position $\boldsymbol{R}_{{c}}$ after the initial acceleration phase. It is a good measure for the position of the blob front. The COM position $\boldsymbol{R}_{{c}}$ and the position of the maximal electron density amplitude $\boldsymbol{R}_{{\text{max}}}$ are visualised in figures 2 and 3. The displacement of the maximal electron density amplitude $\boldsymbol{R}_{{\text{max}}}$ is defined by $\Delta \boldsymbol{R}_{{\text{max}}}(t)\equiv \boldsymbol{R}_{{\text{max}}}(t)-\boldsymbol{R}_{{\text{max}}}(0)={{\left(\Delta {{X}_{\text{max}}}(t), \Delta {{Y}_{\text{max}}}(t)\right)}^{T}}$.

3.2.1. Cold ions.

As illustrated in section 3.1 the COM motion of cold ion temperature (${{\tau}_{\text{i}}}=0$) blobs is purely radial before the up–down symmetry is broken. For ${{\tau}_{\text{i}}}=0$ differences in the propagation of isothermal and thermal blobs should be described by our scaling, which we used to derive the inertial velocity scaling law of equation (28). Figure 5(a) shows that the radial COM displacement $\Delta {{X}_{c}}$ for a given time t is increased by the initial amplitude A. In order to quantify the amplitude scaling of the radial COM displacement $\Delta {{X}_{c}}$, we normalise the radial COM displacement $\Delta {{X}_{c}}$ by the effective blob size ${{\sigma}_{g}}$ and the time by the ideal interchange rate ${{\gamma}_{g}}$. This must then result in nearly overlapping plots if the velocity scaling estimate of equation (28) is correct. This is shown in figure 5(b) and reveals that the radial COM velocities obey the velocity scaling estimate of equation (28) in the cold ion limit. Figure 5(b) also shows that in the initial phase the radial displacement experiences a constant acceleration phase with $\Delta {{X}_{c}}(t)\sim {{V}_{\bot}}{{\gamma}_{g}}{{t}^{2}}$, which is followed by a constant velocity phase with $\Delta {{X}_{c}}(t)\sim {{V}_{\bot}}t$. Hence, the time integrated radial particle transport depends in both time phases on the initial electron pressure perturbation $\Delta {{p}_{\text{e}}}$. The initial electron pressure amplitude is proportional to $\Delta {{p}_{\text{e}}}\sim {{n}_{\text{e}0}} \Delta {{t}_{\text{e}}}+{{t}_{\text{e}0}} \Delta {{n}_{\text{e}}}+ \Delta {{n}_{\text{e}}} \Delta {{t}_{\text{e}}}$ for a thermal blob whereas it scales like $\Delta {{p}_{\text{e}}}\sim {{t}_{\text{e}0}} \Delta {{n}_{\text{e}}}$ for an isothermal blob. Consequently, thermal blobs travel faster radially than isothermal blobs as depicted by figure 5(a). We discuss the implications of the COM displacement $\Delta \boldsymbol{R}_{{c}}$ for the (non time integrated) perpendicular particle transport $\boldsymbol{}{{\mathbf{\Gamma}}_{{{n}_{\text{e}}}}}$ in section 3.4.

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3.2.2. Hot ions.

Finite ion temperature effects (${{\tau}_{\text{i}}}>0$) redistribute the total COM momentum into the radial and poloidal directions. An ion temperature perturbation $\Delta {{t}_{\text{i}}}$ enhances the total (radial and poloidal) COM displacement $| \Delta \boldsymbol{R}_{{c}}|\equiv \sqrt{ \Delta X_{c}^{2}+ \Delta Y_{c}^{2}}$ compared to isothermal blobs. This behaviour is shown for the radial COM displacement $\Delta {{X}_{c}}$ in figure 6(a). The normalisation by the effective cross-field size ${{\sigma}_{g}}$ and the ideal interchange rate ${{\gamma}_{g}}$ is depicted in figure 6(b). This figure shows that the normalised radial COM displacement $\Delta {{X}_{c}}/{{\sigma}_{g}}$ of isothermal and thermal blobs nearly coincide in the constant acceleration and velocity phases. Hence, the radial COM velocities V_{c, x} are captured by the inertial velocity scaling estimate ${{V}_{\bot}}$ of equation (28). Since this estimate is only suitable for the radial COM velocities, modifications for total and poloidal velocity estimates are required. In section 3.4 we show that the total and poloidal velocity scaling laws can be adjusted with the help of the FLR strength parameter $\Theta$.

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In the following we track the radial and poloidal COM position $\boldsymbol{R}_{{c}}$ and investigate the blob propagation for the complete parameter space.

In figure 7 we clarify the dependence of the COM displacement $\boldsymbol{R}_{{c}}$ on the main parameters ($\sigma,{{\tau}_{\text{i}}},A$) for isothermal and thermal blobs. We observe that the radial and poloidal COM displacement $\Delta {{X}_{c}}$ and $\Delta {{Y}_{c}}$ increases with amplitude A over the complete parameter space. This is in accordance with the scaling estimates of the radial COM displacement $\Delta {{X}_{c}}$, presented in section 3.2.1, and holds similarly for the total COM displacement $| \Delta \boldsymbol{R}_{{c}}|$. The poloidal COM displacement $\Delta {{Y}_{c}}$ is decreased by the cross-field size σ and is also enhanced by the ratio of ion to electron background temperature ${{\tau}_{\text{i}}}$. In addition, a finite ion temperature perturbation $\Delta {{t}_{\text{i}}}$ enhances the poloidal COM displacement $\Delta {{Y}_{c}}$ especially for high amplitudes A in comparison to isothermal blobs with constant ion temperature t_i0. This is in line with the FLR effect strength parameter $\Theta$ of equation (29) and is further discussed in section 3.4.

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We ascribe the small increase of poloidal motion to dynamic FLR effects. We do so because a finite ion temperature perturbation $\Delta {{t}_{\text{i}}}$ together with dynamic FLR effects ${{\rho}_{\text{i}}}\left({{T}_{\text{i}}},B\right)$ result in the FLR effect strength parameter $\Theta \sim \sqrt{ \Delta p_{\text{i}}^{2}/\left(\left(\Delta {{p}_{\text{e}}}+ \Delta {{p}_{\text{i}}}\right){{p}_{\text{e}0}}\right)}$. This exceeds its isothermal, constant FLR effect limit equivalent $\Theta \sim \sqrt{t_{\text{i}0}^{2} \Delta {{n}_{\text{e}}}/\left(\left({{t}_{\text{e}0}}+{{t}_{\text{i}0}}\right){{p}_{\text{e}0}}\right)}$ for all studied parameters. This difference stems from either constant or dynamic FLR effects in the polarisation equation (1). In case of ion temperature variations $\Delta {{t}_{\text{i}}}$ dynamic FLR effects appear in the polarisation equation (1) via the gyroaveraging operator $\Gamma _{\text{i},1}^{\dagger}$ whereas in the isothermal limit constant FLR effects enter the polarisation equation (1) via the gyroaveraging operator ${{ \Gamma }_{\text{i},1}}$. In order to ensure that the poloidal motion depends on constant or dynamic FLR effects, we repeated the simulations with temperature dynamics $\left({{t}_{\text{e}}},{{T}_{\text{i}}}\right)\ne \left({{t}_{\text{e}0}},{{T}_{\text{i}0}}\right)$ but constant FLR effects ${{\rho}_{\text{i}}}\left({{T}_{\text{i}0}},{{B}_{0}}\right)$. The related numerical results and the influence of FLR effects on the poloidal particle transport of a blob is presented in section 3.4.

The ability of a blob to retain its initial shape is given by the blob compactness ${{I}_{\text{C}}}(t)$ [27]

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{I}_{\text{C}}}(t)\equiv \frac{{\int}^{}\text{d}\boldsymbol{x}\left({{n}_{\text{e}}}\left(\boldsymbol{x},t\right)-{{n}_{\text{e}0}}\right)h\left(\boldsymbol{x},t\right)}{{\int}^{}\text{d}\boldsymbol{x}\left({{n}_{\text{e}}}\left(\boldsymbol{x},0\right)-{{n}_{\text{e}0}}\right)h\left(\boldsymbol{x},0\right)}, \end{array}\end{eqnarray} \tag{ 33 }$$

with the Heaviside function

$$\begin{eqnarray}h\left(\boldsymbol{x},t\right)=\left\{\begin{array}{*{35}{l}} 1 & ~\text{for}\parallel \boldsymbol{x}-\boldsymbol{R}_{{\text{max}}}(t){{\parallel}^{2}}<{{\sigma}^{2}} \\ 0 & ~\text{else} \end{array}\right. \end{eqnarray} \tag{ 34 }$$

and with $\boldsymbol{R}_{{\text{max}}}(t)$ defined as the position of the maximum electron density amplitude. The lowering of compactness is accompanied by the loss of mass via collisional dissipation and turbulent mixing and stretching whereas a finite Larmor radius contributes to long lived coherent compact structures. Consequently, from figures 2 and 3 we expect low compactness for cold blobs and high compactness for hot blobs.

The maximal radial particle transport of a blob is determined by the blob compactness and the maximal radial velocity. As a result we plot the blob compactness at the time $t=4\gamma _{g}^{-1}$, which approximately coincides with the first maximum of the radial velocity, versus the FLR effect strength $\Theta$ (see equation (29)) in figure 8. Note that for high ${{\tau}_{\text{i}}}$ the maximal radial velocity is often reached at the second peak in time (see e.g. figure 11), which is accompanied by a small drop in compactness I_C. In figure 8 a smooth transition to highly compact blobs is obtained at $\Theta \approx 1$. Above this threshold typically only 10 percent of the initial particle density is lost whereas below the threshold up to 50 percent of the initial particle density is dissipated. For constant ${{\tau}_{\text{i}}}$ the transition to the ion diamagnetic vorticity dominated regime ($\Theta \gg 1$) is reached by decreasing the cross-field size σ or increasing the amplitude of the blob. The thermal and isothermal blobs approach similar values of compactness I_C over the complete $\Theta$ range. We note here that for the sake of clarity the zero ion temperature value ${{\tau}_{\text{i}}}=0$ was replaced by ${{\tau}_{\text{i}}}=0.01$ in order to compute a finite FLR strength $\Theta$.

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On the other hand, the FLR strength parameter $\Theta$ of thermal blobs is larger than of isothermal blobs for constant cross-field size σ, amplitude A and ion to electron background ratio ${{\tau}_{\text{i}}}$. As an example for $\sigma =10$, A  =  0.5 and ${{\tau}_{\text{i}}}=2$ the FLR strength parameter is $\Theta \approx 5$ for thermal blobs and $\Theta \approx 3$ for isothermal blobs. Consequently, figure 8 shows that thermal blobs remain more compact than isothermal blobs. This is underlined by figure 9, which reveals that thermal blobs with dynamic FLR effects lose less mass than thermal or isothermal blobs with constant FLR effects. This is due to differences in the polarisation charge density for constant and dynamic FLR effects, which is discussed in section 2.1. Since the polarisation charge density is associated to the $\boldsymbol{E}\times \boldsymbol{B}$ vorticity, dynamic FLR effects lead to stronger $\boldsymbol{E}\times \boldsymbol{B}$ shear flows in the blob edge in comparison to constant FLR effects. These $\boldsymbol{E}\times \boldsymbol{B}$ shear flows help to retain the initial blob shape (see section 3.1). As a result the compactness of blobs with a dynamic gyro-radius is higher than of blobs with a constant gyro-radius.

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In the following we use the COM velocity $\boldsymbol{V}_{{c}}$ as a measure for the perpendicular particle transport

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \boldsymbol{}{{\mathbf{\Gamma}}_{{{n}_{\text{e}}}}}(t)\approx M\boldsymbol{V}_{{c}}. \end{array}\end{eqnarray} \tag{ 35 }$$

The COM velocity is given by:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \boldsymbol{V}_{{c}}\equiv \frac{\text{d}\boldsymbol{R}_{{c}}}{\text{d}t}. \end{array}\end{eqnarray} \tag{ 36 }$$

In the remainder of this section the influence of the ratio of ion to electron background temperature ${{\tau}_{\text{i}}}$, cross-field size σ and initial amplitude A on the transport of isothermal and thermal blobs is studied.

3.4.1. Cold ions.

In case of zero ion temperature the radial and total transport coincide. In figure 10 the maximal COM velocity is reached after an initial constant acceleration phase and is captured within twenty percent by the inertial velocity scaling law ${{V}_{\bot}}$ of equation (28). The time period to reach the maximal COM velocity scales with the global ideal interchange rate ${{\gamma}_{g}}$ of equation (27).

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3.4.2. Hot ions.

Finite ion temperature induces a poloidal COM velocity in the $\widehat{\boldsymbol{b}}\times \boldsymbol{}\nabla B$ direction. The poloidal COM velocity V_{c, y} is generated by FLR effects and is further decreased (see figure 7) if dynamic FLR effects arise as a consequence of gradients in ion temperature t_i or magnetic field B. Consequently, in figure 11(b) thermal blobs with full FLR dynamics show increased absolute values for the normalised minimal poloidal COM velocities $\text{min}\left({{V}_{c,y}}\right)/{{V}_{\bot}}$ in comparison to blobs with constant FLR effects especially for high amplitudes.

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The behaviour of the normalised radial COM velocities $\text{max}\left({{V}_{c,x}}\right)/{{V}_{\bot}}$ of thermal blobs is similar to blobs with constant FLR effects as depicted in figure 11(a). The maximal normalised radial COM velocities $\text{max}\left({{V}_{c,x}}\right)/{{V}_{\bot}}$ are well described by the velocity scaling estimates of equation (28). As in the cold ion case the maximal radial COM velocities $\text{max}\left({{V}_{c,x}}\right)$ of thermal blobs exceed those of isothermal blobs, which is shown in figure 12. However, the ion temperature scaling for the maximal radial and total COM velocity (see figure 13) is in line with the theoretical estimate ${{V}_{\bot}}\sim \sqrt{1+{{\tau}_{\text{i}}}}$ of equation (28) for blobs with high cross-field size $\sigma =20$. For smaller blob sizes $\sigma =5$ and $\sigma =10$ we observe approximately a decreased scaling ${{V}_{\bot}}\sim {{\left(1+{{\tau}_{\text{i}}}\right)}^{3/8}}$ and ${{V}_{\bot}}\sim {{\left(1+{{\tau}_{\text{i}}}\right)}^{5/12}}$, respectively.

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In figure 14 the time dependence of the normalised total COM velocities ${{V}_{c}}/{{V}_{\bot}}$ are plotted. The maximal total COM velocities V_c agree with the velocity scaling estimates of equation (28) but are slightly underestimated by the velocity scaling estimates of equation (28) in case of dynamic FLR effects. The time period to reach this maximum is enhanced for high blob amplitudes A.

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To verify the inertial velocity scaling law we plot the measured maximal radial COM velocities $\text{max}\left({{V}_{c,x}}\right)$ against the analytical estimates ${{V}_{\bot}}$ of equation (28). Before we do so we note that all the measured velocities are in the inertial regime. The ion pressure dominated RB (iRB) velocity estimate of [12] is not confirmed by our simulations even if some of the parameters fall into this regime. The iRB velocity underestimates the numerically obtained maximal COM velocities, which is in accordance with [50]. In figure 15 we prove numerically that the theoretical velocity estimate of the inertial regime (see equation (28)) is recovered over more than an order of magnitude for the thermal and isothermal blobs. All the derived maximal radial COM velocities lie within 20 percent of the ${{V}_{\bot}}$ estimate, which underlines the usefulness and quality of the inertial scaling law. We tested also the global amplitude velocity scaling ${{V}_{\bot,g}}\sim \sqrt{\left(\Delta {{p}_{\text{e}}}+ \Delta {{p}_{\text{i}}}\right)/\left({{n}_{\text{e}0}}+ \Delta {{n}_{\text{e}}}\right)}$ of [28, 45]. However, we obtained distinct deviations from the numerically obtained maximal COM velocities $\text{max}\left({{V}_{c,x}}\right)$ by the global inertial scaling law ${{V}_{\bot,g}}$. Hence, we emphasise that the inertial velocity scaling law ${{V}_{\bot}}$ of equation (28) captures the maximal radial COM velocities $\text{max}\left({{V}_{c,x}}\right)$ best.

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In sections 3.2-3.4 we discussed the parameter $\Theta$ defined in equation (29), which is a measure of the influence of FLR effects on blob convection. Specifically we argued that the poloidal and consequently the total transport is related to the FLR strength parameter $\Theta$. We will now derive an empirical scaling law for the poloidal and total particle transport based on the knowledge of our parameter study. The tracks of the COM position $\boldsymbol{R}_{{c}}$ in figure 7 indicate that for high FLR strength (equation (29)) an angle of $-\pi /4$ is taken by the blobs. Hence, we postulate that this angle is approached by

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \alpha \equiv \frac{1}{2}\frac{{{\tan}^{-1}}\left(\Theta +{{ \Theta }_{s}}\right)-{{\tan}^{-1}}\left({{ \Theta }_{s}}\right)}{1-2{{\tan}^{-1}}\left({{ \Theta }_{s}}\right)/\pi}, \end{array}\end{eqnarray} \tag{ 37 }$$

which is a shifted ${{\tan}^{-1}}$ function with shift parameter ${{ \Theta }_{s}}$. The angle α captures the blob compactness ${{I}_{c}}\left(\alpha \right)={{c}_{1}}\alpha +{{c}_{2}}$ at $t=4\gamma _{g}^{-1}$ up to the constants c₁ and c₂. We find that the constants ${{c}_{1}}=\frac{\pi}{2}$, ${{c}_{2}}=\frac{1}{2}$ and the shift parameter ${{\theta}_{s}}=5$ fit the blob compactness ${{I}_{\text{C}}}\left(4\gamma _{g}^{-1}\right)$ with a threshold of $4{{\sigma}^{2}}$ (see equation (34)). With equation (37) we derive the total and poloidal velocity scaling law

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} V\equiv {{V}_{\bot}}/\cos \left(\alpha \right), \end{array}\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{V}_{y}}\equiv {{V}_{\bot}}\tan \left(\alpha \right). \end{array}\end{eqnarray} \tag{ 39 }$$

In figures 16 and 17 we compare the derived total and poloidal inertial scaling law V of equation (38) and V_y of equation (39) with the numerically obtained maximal total COM velocities $\text{max}\left({{V}_{c}}\right)$ and the numerically obtained minimal poloidal COM velocities $\text{min}\left({{V}_{c,y}}\right)$. For the total blob velocities we find again agreement up to $\pm 20$ percent. On the other hand, the minimal poloidal COM velocities of the thermal and isothermal blobs are not always within 20 percent of the theoretical estimate of equation (39). This occurs especially for very small poloidal velocities. Still, figure 17 underlines the quality of the empirical scaling law.

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