Diagnostics

This page lists dynamo output diagnostics common in the literature, how to calculate them from leeds dynamo code outputs, and references to alternative definitions.

On this page, a quantity with a superscript $^*$ indicates it has been non-dimensionalised, so $\chi^*$ is the non-dimensionalised version of $\chi$.

Magnetic Energy

Dimensional magnetic energy is defined as $E_\text{mag} = \frac{1}{2\mu_0}\int{\boldsymbol{B^2}}dV$, where $\mu_0$ is the magnetic permeability. In the dynamo code, energy is non-dimensionalised by $\rho D \eta^2$, where $\rho$ is the density, $D$ is the shell thickness, and $\eta$ is magnetic diffusivity.

Magnetic energy is calculated in the code as

\[ E_\text{mag}^* = \frac{\text{Pm}}{\text{Ek}}\int{(\boldsymbol{B}^*)^2}dV^*, \]

where Pm is the magnetic Prandtl number and Ek is the Ekman number. Can be found in column 2 of mag_energy.dat.

Magnetic Energy Density

Magnetic Energy divided by the shell volume.

\[ E_\text{mag}^{*\text{density}} = \frac{\text{Pm}}{\text{Ek}V^*}\int{(\boldsymbol{B}^*)^2}dV^*. \]

For Earth-like simulations, $E_\text{mag}^{*\text{density}}\approx E^*_\text{mag} / 14.59$.

Kinetic Energy

Dimensional kinetic energy defined as $E_\text{kin} = \frac{1}{2}\rho\int\boldsymbol{u}^2 dV$, where $V$ is the shell volume, and $\rho$ is the fluid density. Energy is non-dimensionalised by $\rho D \eta^2$, where $D$ is the shell thickness, and $\eta$ is magnetic diffusivity. Non-dimensional kinetic energy is calculated in the dynamo code as

\[ E_\text{kin}^* = \frac{1}{2}\int(\boldsymbol{u}^*)^2 dV. \]

It can be found in column 2 of vel_energy.dat.

Kinetic Energy Density

Kinetic Energy divided by the shell volume.

\[ E_\text{kin}^{*\text{density}} = \frac{1}{2V^*}\int (\boldsymbol{u}^*)^2 dV^*. \]

For Earth-like geometry, $E_\text{kin}^{*\text{density}}\approx E^*_\text{kin}/14.59$.

Ohmic Dissipation

Dimensional ohmic dissipation defined as $D_\Omega = \eta/\mu_0\int(\nabla\times\boldsymbol{B})^2dV$, non dimensionalised with $\rho\eta^3/D$ where $\eta$ is magnetic diffusivity, $\mu_0$ is magnetic permeability, $\rho$ is density.

Calculated in code as

\[ D_\Omega^* = \frac{2 \text{Pm}}{\text{Ek}}\int\left(\nabla^* \times \boldsymbol{B}^*\right)^2 dV^*, \]

where Pm is the magnetic Prandtl number and Ek is the Ekman number.

Can be found in column 7 of mag_energy.dat.

Viscous Dissipation

Dimensional viscous dissipation defined as $D_\nu = \rho\nu\int\left(\nabla \times \boldsymbol{u}\right)^2 dV$, non dimensionalised with $\rho\eta^3/D$, where $\rho$ is the density, $\nu$ is the kinematic viscosity, and $\eta$ is the magnetic diffusivity.

Calculated in code as

\[ D_\nu^* = \text{Pm}\int\left(\nabla^* \times \boldsymbol{u^*}\right)^2 dV^*, \]

where Pm is the magnetic Prandtl number.

Found in column 7 of vel_energy.dat.

Magnetic Reynolds Number

Defined as $$ \text{Rm} = \frac{U D}{\eta}, $$ where $U$ is a characteristic velocity, $D$, the characteristic lengthscale, is taken as the shell thickness, and $\eta$ is the magnetic diffusivity.

In the code, Rm is calculated as

\[ \text{Rm} = \sqrt{\frac{2 E^*_\text{kin}}{V^*}}, \]

where $E^*_\text{kin}$ is the non-dimensional kinetic energy and $V^*$ is the non-dimensional shell volume.

For Earth like geometry, can be plotted from gnuplot using vel_energy.dat as

plot "vel_energy.dat" u 1:(sqrt(2*$2/14.59)) w l

or directly from column 2 of dynamo_outputs.dat.

Reynolds number

Defined as

\[ \text{Re} = \frac{U D}{\nu}, \]

where $U$ is a characteristic velocity, $D$, the characteristic lengthscale, is taken as the shell thickness, and $\nu$ is the kinematic viscosity.

In the code, Re calculated as

\[ Re = \frac{1}{\text{Pm}}\sqrt{\frac{2 E^*_\text{kin}}{V^*}}, \]

where Pm is the magnetic Prandtl number, $E^*_\text{kin}$ is the non-dimensional kinetic energy, and $V^*$ is the non-dimensional shell volume.

For Earth like geometry, can be plotted from gnuplot using vel_energy.dat as

plot "vel_energy.dat" u 1:((1/Pm)*sqrt(2*$2/14.59)) w l

where Pm should be replaced by its numerical value, or directly from column 3 of dynamo_outputs.dat.

Peclet Number

Defined as

\[ \text{Pe} = \frac{U D}{\kappa} \]

where $U$ is a characteristic velocity, $D$, the characteristic lengthscale, is taken as the shell thickness, and $\kappa$ is the thermal diffusivity.

In the code, Pe calculated as

\[ Pe = \frac{Pr}{\text{Pm}}\sqrt{\frac{2 E^*_\text{kin}}{V^*}}, \]

where Pr is the Prandtl number, Pm is the magnetic Prandtl number, $E^*_\text{kin}$ is the non-dimensional kinetic energy, and $V^*$ is the non-dimensional shell volume.

For Earth like geometry, can be plotted from gnuplot using vel_energy.dat as

plot "vel_energy.dat" u 1:((Pr/Pm)*sqrt(2*$2/14.59)) w l

where Pr and Pm should be replaced by their numerical values, or directly from column 4 of dynamo_outputs.dat.

Compositional Peclet Number

Rossby Number

Defined as

\[ \text{Ro} = \frac{U}{\Omega D}, \]

where $U$ is a characteristic velocity, $D$, the characteristic lengthscale, is taken as the shell thickness, and $\Omega$ is the rotation rate.

In the code, Ro calculated as

\[ \text{Ro} = \frac{Ek}{\text{Pm}}\sqrt{\frac{2 E^*_\text{kin}}{V^*}}, \]

where Ek is the Ekman number, Pm is the magnetic Prandtl number, $E^*_\text{kin}$ is the non-dimensional kinetic energy, and $V^*$ is the non-dimensional shell volume.

Can be plotted from gnuplot using vel_energy.dat as

plot "vel_energy.dat" u 1:((Ek/Pm)*sqrt(2*$2/V*)) w l

where Ek, Pm, and $V^*$ should be replaced by their numerical values, or directly from column 6 of dynamo_outputs.dat.

M_Ratio

Defined as

\[ M = \frac{E_\text{mag}}{E_\text{kin}}, \]

where $E_\text{mag}$ is the dimensional magnetic Energy and $E_\text{kin}$ is the dimensional kinetic Energy.

Calculated in the code as

\[ M = \frac{E^*_\text{mag}}{E^*_\text{kin}}, \]

where $E_\text{mag}^*$ and $E_\text{kin}^*$ are the non-dimensional magnetic and kinetic energies.

Can be plotted from gnuplot using vel_energy.dat and mag_energy.dat as

plot '< paste mag_energy.dat vel_energy.dat' using 1:($2/$8) w l

or from column 7 of dynamo_outputs.dat.

A related qunatitiy is the Squared Alfvén number

Elsasser Number

There are two definitions of the Elsasser number common in the literature. Soderlund et al. (2012) and others define

\[ \Lambda = \frac{\boldsymbol{B}^2}{2\Omega\rho\mu_0\eta}, \]

where $\boldsymbol{B}$ is the dimesional magnetic field, $\Omega$ is the rotation rate, $\rho$ is the fluid density, $\mu_0$ is the magnetic permeability, and $\eta$ is the magnetic diffusivity.

Meanwhile, Aubert (2017, 2019) etc. define

\[ \Lambda_\text{A17} = \frac{\boldsymbol{B}^2}{\Omega\rho\mu_0\eta}. \]

$\Lambda$ as in Soderlund et al. (2012) is calculated in the dynamo code as

\[ \Lambda = \frac{\text{Ek}E^*_\text{mag}}{\text{Pm} V^*}, \]

Where Ek is the Ekman number, $E^*_\text{mag}$ is the non-dimensional magnetic energy, Pm is the magnetic Prandtl number, and $V^*$ is the non-dimensional shell volume.

$\Lambda$ is written out to column 8 of dynamo_outputs.dat. To convert to $\Lambda_\text{A17}$, multiply by 2.

$\Lambda$ can also be plotted in gnuplot using mag_energy.dat as

p "mag_energy.dat" using 1:($2*Ek/(V* *Pm)) w l,

where Ek, $V^*$ and Pm should be replaced by their numerical values.

Lehnert Number

We define the Lehnert number as in Aubert (2017, 2019) etc.

\[ \lambda = \frac{B}{\left(\rho\mu_0\right)^{1/2}\Omega D}, \]

where $B$ the magnetic field magnitude, $\rho$ is the fluid density, $\mu_0$ is the magnetic permeability, $\Omega$ is the rotation rate, and $D$ is the shell thickness.

Calculated in the dynamo code as

\[ \lambda = \frac{\text{Ek}}{\text{Pm}}\sqrt{\frac{2 E_\text{mag}^*}{V^*}}, \]

where Ek is the Ekman number, Pm is the magnetic Prandtl number, $E_\text{mag}^*$ is the non-dimensional magnetic energy, and $V^*$ is the non-dimensional shell volume.

Written out to column 9 of dynamo_outputs.dat.

Can also be calculated using mag_energy.dat, for example in gnuplot

p "mag_energy.dat" using 1:((Ek/Pm)*sqrt(2*$2/V*)) w l

where Ek, Pm, and $V^*$ should be replaced by their numerical values.

fdip

Fraction of dipolarity of CMB magnetic field, defined as in Christensen and Aubert (2006)

\[ \mathcal{f}_\text{dip} = \left( \frac {\int{ \boldsymbol{B}_{l=1}(r=r_o) \cdot \boldsymbol{B}_{l=1}(r=r_0) }dS } {\int{ \boldsymbol{B}_{l\leq 12}(r=r_o) \cdot \boldsymbol{B}_{l\leq 12}(r=r_0) }dS} \right)^{1/2}, \]

where $\boldsymbol{B}_{l}(r)$ is the magnetic field at spherical harmonic degrees $l$ and radial position $r_o$, and $S$ is the spherical surface. Due to this being a ratio of field strengths, no conversions are required due to choice of non-dimensionalisation.

$\mathcal{f}_\text{dip}$ is written out to column 10 of dynamo_outputs.dat.

Can also be calculated using mag_cmb.dat, for example in gnuplot

p "mag_cmb.dat" u 1:(sqrt($4/$8)),

where column 4 is the surface magnetic energy at $l=1$ and column 8 is the surface magnetic energy for $l\leq 12$.

The comparison between the dipole and the $l\leq 12$ surface magnetic field is made because of comparisons with observational data of the Earth's magnetic field, which is limited in resolution to $l\approx 13$ due to the crustal magnetic field.

Some literature uses the comparison between the dipole and the whole of the rest of the field

\[ \left( \frac {\int{ \boldsymbol{B}_{l=1}(r=r_o) \cdot \boldsymbol{B}_{l=1}(r=r_0) }dS } {\int{ \boldsymbol{B}(r=r_o) \cdot \boldsymbol{B}(r=r_0) }dS} \right)^{1/2}, \]

which can be calculated in the dynamo code using columns 4 and 2 of mag_cmb.dat.

fohm

Fraction of ohmic dissipation. Defined as

\[ f_\text{ohm} = \frac{D_\Omega}{D_\Omega + D_\nu} \]

where $D_\Omega$ is the ohmic dissipation and $D_\nu$ is the viscous dissipation.

Calculated in the code as

\[ f_\text{ohm} = \frac{D^*_\Omega}{D^*_\Omega + D^*_\nu}, \]

and written out to column 11 of dynamo_outputs.dat.

Can also be calculated using vel_energy.dat and mag_energy.dat, for example in gnuplot

p "< paste mag_energy.dat vel_energy.dat" u 1:($6/($6+$13)) w l

Theta_dip

The dipole tilt angle, defined as the angle between the magnetic field dipole and the rotation axis at the north pole. Defined as

\[ \theta = \arccos\left(-\frac{M_z}{|\boldsymbol{M}|} \right) \]

where $\theta$ is the dipole tilt angle, $M_z$ is the $\hat{\boldsymbol{z}}$ component of the magnetic dipole moment, $|\boldsymbol{M}|$ is the dipole magnitude.

Calculated as $$ \theta = \arccos\left(\frac{g_1^0}{g_1}\right) $$

where $g_1=\sqrt{\left(g_1^1\right)^2 + \left(h_1^1\right)^2 + \left(g_1^0\right)^2}$ is the non-dimensional dipole magnitude, and $g_l^m$, $h_l^m$ are the gauss coefficients of degree $l$ and order $m$.

Due to the poloidal-toridal decomposition, the dynamo code takes $g_1^0 = B^\text{pol}_{1,0}$, $~g_1^1=-2\Re(B^\text{pol}_{1,1})$, and $h_1^1=2\Im(B^\text{pol}_{1,1})$, where $B^\text{pol}_{l,m}$ is the poloidal magnetic scaler of degree $l$ and order $m$, and $\Re$ and $\Im$ are the real and imaginary coefficients, resepectively.

Written out to column 12 of dynamo_outputs.dat, and column 5 of mag_cmb.dat.

Nusselt

written out to column 13 of dynamo_outputs.dat.

Sherwood

written out to column 14 of dynamo_outputs.dat.