Temperature Profile | ElectronTemperatureProfile(Profile)

The temperature profile class is organised around a central runner function that is called each time the plasma is parameterised by the parent PlasmaProfile() class. It is called by pedestal_parameterisation() and parabolic parameterisation(). The sequence of the runner function can be seen below along with explanation of the following calculations.

Runner function | run()

1. Firstly the profile x-dimension is normalised in normalise_profile_x() by simply dividing the profile size by its max value.

2. The steps between the normalised points is then done by calculate_profile_dx() which divides the max x-dimension by the number of points.

3. The core electron and ion temperatures are claculated via set_physics_variables()

Calculate core values | set_physics_variables()

Electron core temperature of a pedestalised profile | calculate_pedestal_profile_on_axis_temperature()

This function calculates the core electron temperature for a pedestalsied profile in $\text{keV}$. The inclusion of a new $\beta_T$ exponent term allows a more accurate description of temperature profiles with a triangular shape or a strong gradient near the pedestal (characteristic of regimes with an internal transport barrier).

A list of input parameters for calculating the core plasma temperature can be found below.

Profile parameter / Input	Temperature
Pedestal radius (r/a)	radius_plasma_pedestal_temp_norm, $\rho_{\text{ped,T}}$
Pedestal value	temp_plasma_pedestal_kev, $T_{\text{ped}}$
Separatrix value	temp_plasma_separatrix_kev, $T_{\text{sep}}$
Average temperature	temp_vol_avg_kev, $\langle T \rangle$
Profile index/ peaking parameter	alphat, $\alpha_T$
Profile index/ peaking parameter	tbeta, $\beta_T$

$$T_0 = T_{\text{ped}}+\frac{\beta_T(3\langle T_{\text{e}} \rangle +T_{\text{sep}}(-2+\rho_{\text{ped}} +\rho_{\text{ped}}^2)-T_{\text{ped}}(1+\rho_{\text{ped}}+\rho_{\text{ped}}^2))}{6\rho_{\text{ped}}^2\text{B}\left[1+\alpha_T,\frac{2}{\beta_T}\right]}$$

Where $\text{B}$ is the Beta function

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Derivation

We calculate the volume integrated profile and then divide by the volume of integration to get the volume average temperature $\langle T_{\text{e}} \rangle$. If we assume the plasma to be a torus of circular cross-section then we can use spherical coordinates. We can simplify the problem by representing the torus as a cylinder of height equal to the circumference of the torus, which is equal to $2\pi R$, where $R$ is the major radius of the torus, and $a$ is the plasma minor radius in the poloidal plane.

The cylindrical volume element is given by:

$$V = \int \int \int dV = \int^{2\pi R}_0 \int^{2\pi}_0 \int^a_0 r \ dr \ d\theta \ dz$$

Inserting our temperature function to be integrated over we get in normalised radial coordinates $(\rho)$ we get:

$$\int^{2\pi R}_0 \int^{2\pi}_0 \int^{1}_0 \rho \ T_{\text{e}}(\rho) \ d\rho \ d\theta \ dz$$

Since our temperature function is only a function of $\rho$, and the torus is symmetric around its center, the integration simplifies to integrating over $\rho$ and the $d\theta ,\ dz$ integrals are solved to give values for the full poloidal angle and cylindrical height / torus length, leading to:

$$4\pi^2R \int^{1}_0 \rho \ T_{\text{e}}(\rho) \ d\rho$$

This is the general form for the full profile width without expansion. Separating out the temperature function into its separate functions for the core and pedestal region we get the fully expanded integration form.

$$4\pi^2R\left[ \int^{\rho_{\text{ped,T}}}_0 \rho\left(T_{\text{ped}} + (T_0 - T_{\text{ped}}) \left( 1 - \frac{\rho^{\beta_T}}{\rho_{\text{ped},T}^{\beta_T}}\right)^{\alpha_T}\right) \ d\rho \\ +\int^1_{\rho_{\text{ped,T}}} \rho\left(T_{\text{sep}} + (T_{\text{ped}} - T_{\text{sep}})\left( \frac{1- \rho}{1-\rho_{\text{ped},T}}\right)\right)\right] \ d\rho$$

In the form of volume average temperature where the volume integrated temperature function has to be divided by the volume of the cylinder / torus, within the volume bounded by that pedestal position we get:

$$\langle T_{\text{e}} \rangle = 4\pi^2R\left[ \frac{\frac{\left(T_{\text{ped}}\beta_T+(2T_0-2T_{\text{ped}})B\left(\alpha_T+1,\frac{2}{\beta_T}\right)\right)\rho_{\text{ped},T}^2}{2\beta_T}+\frac{(1-\rho_{\text{ped},T})\left((T_{\text{sep}}+2T_{\text{ped}})\rho_\text{ped}+2T_{\text{sep}}+T_{\text{ped}}\right)}{6}}{2\pi^2 R \rho^2}\right]$$

In this case, the value of $\rho$ is equal to 1 as we integrated over the full profile.

$$\langle T_{\text{e}} \rangle = 2\left[ \frac{\left(T_{\text{ped}}\beta_T+(2T_0-2T_{\text{ped}})\text{B}\left(\alpha_T+1,\frac{2}{\beta_T}\right)\right)\rho_{\text{ped},T}^2}{2\beta_T} \\ +\frac{(1-\rho_{\text{ped},T})\left((T_{\text{sep}}+2T_{\text{ped}})\rho_\text{ped}+2T_{\text{sep}}+T_{\text{ped}}\right)}{6}\right]$$

$$\langle T_{\text{e}} \rangle = \frac{\left(T_{\text{ped}}\beta_T+(2T_0-2T_{\text{ped}})\text{B}\left[\alpha_T+1,\frac{2}{\beta_T}\right]\right)\rho_{\text{ped},T}^2}{\beta_T} \\ +\frac{(1-\rho_{\text{ped},T})\left((T_{\text{sep}}+2T_{\text{ped}})\rho_\text{ped}+2T_{\text{sep}}+T_{\text{ped}}\right)}{3}$$

Where $\text{B}$ is the Beta function

Re-arranging to get $T_0$ we get:

$$T_0 = T_{\text{ped}}+\frac{\beta_T(3\langle T_{\text{e}} \rangle +T_{\text{sep}}(-2+\rho_{\text{ped}} +\rho_{\text{ped}}^2)-T_{\text{ped}}(1+\rho_{\text{ped}}+\rho_{\text{ped}}^2))}{6\rho_{\text{ped}}^2\text{B}\left[1+\alpha_T,\frac{2}{\beta_T}\right]}$$

$\blacksquare$

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Electron core temperature of a parabolic profile | calculate_parabolic_profile_on_axis_temperature()

This function calculates the core electron temperature for a parabolic profile in $\text{keV}$.

A list of input parameters for calculating the core plasma temperature can be found below.

Profile parameter / Input	Temperature
Average temperature	temp_vol_avg_kev, $\langle T \rangle$
Profile index/ peaking parameter	alphat, $\alpha_T$

$$T_0 = \langle T \rangle \times \left(\alpha_T+1\right)$$

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Derivation

We calculate the volume integrated profile and then divide by the volume of integration to get the volume average temperature $\langle T \rangle$. If we assume the plasma to be a torus of circular cross-section then we can use spherical coordinates. We can simplify the problem by representing the torus as a cylinder of height equal to the circumference of the torus, which is equal to $2\pi R$, where $R$ is the major radius of the torus.

The cylindrical volume element is given by:

$$V = \int \int \int dV = \int^{2\pi R}_0 \int^{2\pi}_0 \int^a_0 r \ dr \ d\theta \ dz$$

Inserting our temperature function in the form where $\rho$ is expanded as $\rho = r/a$ we get:

$$\int^{2\pi R}_0 \int^{2\pi}_0 \int^a_0 r \left(T_0(1-r^2/a^2)^{\alpha_T}\right) \ dr \ d\theta \ dz$$

Since our temperature function is only a function of $r$, and the torus is symmetric around its center, the integration simplifies to integrating over $r$, with the $d\theta$ and $dz$ integrals solving to give values for the full poloidal angle, and cylindrical height and torus length, respectively. This leads to:

$$4\pi^2R \int^a_0 r \left(T_0(1-r^2/a^2)^{\alpha_T}\right) \ dr$$

In the form of volume average temperature where the volume integrated temperature function has to be divided by the volume of the cylinder / torus we get:

$$\langle n \rangle = \frac{4\pi^2R \int^a_0 r \left(T_0(1-r^2/a^2)^{\alpha_T}\right) \ dr}{2\pi^2Ra^2}$$

Simplify:

$$\langle n \rangle = \frac{2 \int^a_0 r \left(T_0(1-r^2/a^2)^{\alpha_T}\right) \ dr}{a^2}$$

Integrate:

$$\langle T \rangle = \frac{2}{a^2} \frac{a^2 T_0}{2(\alpha_T+1)}$$

$$\therefore \langle T \rangle = \frac{T_0}{\alpha_T+1}$$

This is in agreement with the derivation from the ITER Physics Design 1989².

Since all parabolic profiles are of the same form this proof holds for the density and temperature.

A similar derivation is found in normalised coordinates, $\rho$ if the line averaged integral is swept around the poloidal circumference then divided by the poloidal area.

$$\langle T \rangle = \frac{2\pi \int^1_0 \rho \left(T_0(1-\rho^2)^{\alpha_T}\right) \ d\rho}{\pi\rho^2}$$

$\blacksquare$

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The core ion temperature is then set such as:

$$T_{\text{i0}} = \left(\frac{T_{\text{i}}}{T_{\text{e}}}\right)T_{\text{e0}}$$

1. The y profile is then calculated using calculate_profile_y(). This routine calculates the temperature at each normalised minor radius position $\rho$ for a HELIOS-type temperature pedestal profile¹

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Calculate temperature at each radius position | calculate_profile_y()

A table of the input variables can be found below

Profile parameter / Input	Density
Normalised plasma radii	profile_x
Pedestal radius (r/a)	radius_plasma_pedestal_temp_norm, $\rho_{\text{ped,T}}$
Core temperature	temp_plasma_electron_on_axis_kev, $T_{\text{e0}}$
Pedestal value	temp_plasma_pedestal_kev, $T_{\text{ped}}$
Separatrix value	temp_plasma_separatrix_kev, $T_{\text{sep}}$
Profile index/ peaking parameter	alphat, $\alpha_T$
2nd profile index/ peaking parameter	tbeta, $\beta_T$

If i_plasma_pedestal == 0 then the original parabolic profile form is used

$$T(\rho) = T_0(1 - \rho^2)^{\alpha_T}$$

The central temperature ($T_0$) is then checked to make sure it is not less than the pedestal temperature, $T_{\text{ped}}$. If it is less than a logger warning is pushed to the terminal at runtime.

Values of the profile temperature are then assigned based on the temperature function below across bounds from 0 to radius_plasma_pedestal_temp_norm and radius_plasma_pedestal_temp_norm to 1.

$$\begin{aligned} \mbox{Temperature:} \ \ T(\rho) = \left\{ \begin{aligned} & \text{T}_{\text{ped}} + (T_0 - \text{T}_{\text{ped}}) \left( 1 - \frac{\rho^{\beta_T}} {\rho_{\text{ped},T}^{\beta_T}}\right)^{\alpha_T} \ 0 \leq \rho \leq \rho_{\text{ped},T} \\ & \text{T}_{\text{sep}} + (\text{T}_{\text{ped}} - \text{T}_{\text{sep}})\left( \frac{1- \rho}{1-\rho_{\text{ped},T}}\right) \ \rho_{\text{ped},T} < \rho \leq 1 \end{aligned} \right. \end{aligned}$$

1. Profile is then integrated with integrate_profile_y() using Simpsons integration from the profile abstract base class

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1. Jean, J. (2011). HELIOS: A Zero-Dimensional Tool for Next Step and Reactor Studies. Fusion Science and Technology, 59(2), 308–349. https://doi.org/10.13182/FST11-A11650 ↩

2. N.A. Uckan and ITER Physics Group, 'ITER Physics Design Guidelines: 1989', ↩