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<section id="formulation-of-the-magneto-hydrodynamics-problem">
<span id="secequations"></span><h1>Formulation of the (magneto)-hydrodynamics problem<a class="headerlink" href="#formulation-of-the-magneto-hydrodynamics-problem" title="Link to this heading">¶</a></h1>
<p>The general equations describing thermal convection and dynamo action of a
rotating compressible fluid are the starting point from which the Boussinesq or
the anelastic approximations are developed. In MagIC, we consider a spherical
shell rotating about the vertical <span class="math notranslate nohighlight">\(z\)</span> axis with a constant angular
velocity <span class="math notranslate nohighlight">\(\Omega\)</span>. Equations are solve in the corotating system.</p>
<p>The conservation of momentum is formulated by the Navier-Stokes equation</p>
<div class="math notranslate nohighlight" id="equation-eqns">
<span class="eqno">(1)<a class="headerlink" href="#equation-eqns" title="Link to this equation">¶</a></span>\[\rho\left(\dfrac{\partial \vec{u}}{\partial t}+ \vec{u}\cdot\vec{\nabla}\
\vec{u} \right) =-\vec{\nabla} p +
\dfrac{1}{\mu_0}(\vec{\nabla}\times\vec{B})\times\vec{B} +\rho
\vec{g}-2\rho\vec{\Omega}\times\vec{u}+ \vec{\nabla}\cdot\mathsf{S},\]</div>
<p>where <span class="math notranslate nohighlight">\(\vec{u}\)</span> is the velocity field, <span class="math notranslate nohighlight">\(\vec{B}\)</span> the magnetic field,
and <span class="math notranslate nohighlight">\(p\)</span> a modified pressure that includes centrifugal forces.
<span class="math notranslate nohighlight">\(\mathsf{S}\)</span> corresponds to the rate-of-strain tensor given by:</p>
<div class="math notranslate nohighlight">
\[\begin{split}\begin{aligned}
S_{ij} & = 2\nu\rho\left[e_{ij}-\dfrac{1}{3}\delta_{ij}\,\vec{\nabla}\cdot\vec{u} \right], \\
e_{ij} & =\dfrac{1}{2}\left(\dfrac{\partial u_i}{\partial x_j}+\dfrac{\partial
u_j}{\partial x_i}\right).
\end{aligned}\end{split}\]</div>
<p>Convection is driven by buoyancy forces acting on variations in density <span class="math notranslate nohighlight">\(\rho\)</span>.
These variations have a dynamical part formulated by the continuity equation
describing the conservation of mass:</p>
<div class="math notranslate nohighlight" id="equation-eqcontinuity">
<span class="eqno">(2)<a class="headerlink" href="#equation-eqcontinuity" title="Link to this equation">¶</a></span>\[\dfrac{\partial \rho}{\partial t} = - \vec{\nabla} \cdot \rho \vec{u}.\]</div>
<p>In addition an equation of state is required to formulate the thermodynamic
density changes. For example the relation</p>
<div class="math notranslate nohighlight" id="equation-eos1">
<span class="eqno">(3)<a class="headerlink" href="#equation-eos1" title="Link to this equation">¶</a></span>\[\dfrac{1}{\rho} \partial \rho = -\alpha \partial T + \beta \partial p - \delta \partial \xi\]</div>
<p>describes density variations caused by variations in temperature <span class="math notranslate nohighlight">\(T\)</span>,
pressure <span class="math notranslate nohighlight">\(p\)</span>, and composition <span class="math notranslate nohighlight">\(\xi\)</span>. The latter contribution needs
to be considered for describing the effects of light elements released from a
growing solid iron core in a so-called double diffusive approach.</p>
<p>To close the system we also have to formulate the dynamic changes of entropy,
pressure, and composition. The evolution equation for pressure can be derived
from the Navier-Stokes equation, as will be further discussed below. For
entropy variations we use the so-called energy or heat equation</p>
<div class="math notranslate nohighlight" id="equation-eqentropy">
<span class="eqno">(4)<a class="headerlink" href="#equation-eqentropy" title="Link to this equation">¶</a></span>\[\rho T\left(\dfrac{\partial s}{\partial t}+\vec{u}\cdot \vec{\nabla} s \right) =
\vec{\nabla}\cdot (k\vec{\nabla} T) + \Phi_\nu +\dfrac{\lambda}{\mu_0}\left(\vec{\nabla}\times\vec{B}\right)^2
+ \epsilon,\]</div>
<p>where <span class="math notranslate nohighlight">\(\Phi_\nu\)</span> corresponds to the viscous heating expressed by</p>
<div class="math notranslate nohighlight">
\[\Phi_\nu = 2\rho\left[e_{ij}e_{ji}-\dfrac{1}{3}\left(\vec{\nabla}\cdot\vec{u}\right)^2\right]\]</div>
<p>Note that we use here the summation convention over the indices <span class="math notranslate nohighlight">\(i\)</span> and
<span class="math notranslate nohighlight">\(j\)</span>. The second last term on the right hand side is the Ohmic heating due
to electric currents. The last term is a volumetric sink or source term that
can describe various effects, for example radiogenic heating, the mixing-in of
the light elements or, when radially dependent, potential variations in the
adiabatic gradient (see below).
For chemical composition, we finally use</p>
<div class="math notranslate nohighlight" id="equation-eqxi">
<span class="eqno">(5)<a class="headerlink" href="#equation-eqxi" title="Link to this equation">¶</a></span>\[\rho \left(\dfrac{\partial \xi}{\partial t}+\vec{u}\cdot \vec{\nabla} \xi \right) =
\vec{\nabla}\cdot(k_\xi \vec{\nabla} \xi) + \epsilon_\xi,\]</div>
<p>The induction equation is obtained from the Maxwell equations (ignoring
displacement current) and Ohm’s law (neglecting Hall effect):</p>
<div class="math notranslate nohighlight" id="equation-eqinduction">
<span class="eqno">(6)<a class="headerlink" href="#equation-eqinduction" title="Link to this equation">¶</a></span>\[\dfrac{\partial \vec{B}}{\partial t} = \vec{\nabla} \times \left( \vec{u}\times\vec{B}-\lambda\,\vec{\nabla}\times\vec{B}\right).\]</div>
<p>When the magnetic diffusivity <span class="math notranslate nohighlight">\(\lambda\)</span> is homogeneous this simplifies to
the commonly used form</p>
<div class="math notranslate nohighlight" id="equation-eqinduction2">
<span class="eqno">(7)<a class="headerlink" href="#equation-eqinduction2" title="Link to this equation">¶</a></span>\[\dfrac{\partial \vec{B}}{\partial t} = \vec{\nabla} \times \left( \vec{u}\times\vec{B} \right)
+ \lambda\,\vec{\Delta}\vec{B}.\]</div>
<p>The physical properties determining above equations are rotation rate
<span class="math notranslate nohighlight">\(\Omega\)</span>, the kinematic viscosity <span class="math notranslate nohighlight">\(\nu\)</span>, the magnetic permeability
<span class="math notranslate nohighlight">\(\mu_0\)</span>, gravity <span class="math notranslate nohighlight">\(\vec{g}\)</span>, thermal conductivity <span class="math notranslate nohighlight">\(k\)</span>, Fick’s
conductiviy <span class="math notranslate nohighlight">\(k_\xi\)</span>, magnetic diffusivity <span class="math notranslate nohighlight">\(\lambda\)</span>. The latter
connects to the electrical conductivity <span class="math notranslate nohighlight">\(\sigma\)</span> via <span class="math notranslate nohighlight">\(\lambda =
1/(\mu_0\sigma)\)</span>. The thermodynamics properties appearing in <a class="reference internal" href="#equation-eos1">(3)</a> are
the thermal expansivity at constant pressure (and composition)</p>
<div class="math notranslate nohighlight" id="equation-alpha">
<span class="eqno">(8)<a class="headerlink" href="#equation-alpha" title="Link to this equation">¶</a></span>\[\alpha = -\dfrac{1}{\rho}\left(\dfrac{\partial\rho}{\partial T}\right)_{p,\xi},\]</div>
<p>the compressibility at constant temperature</p>
<div class="math notranslate nohighlight">
\[\beta = \dfrac{1}{\rho}\left(\dfrac{\partial\rho}{\partial p}\right)_{T,\xi}\]</div>
<p>and an equivalent parameter <span class="math notranslate nohighlight">\(\delta\)</span> for the dependence of
density on composition:</p>
<div class="math notranslate nohighlight" id="equation-delta">
<span class="eqno">(9)<a class="headerlink" href="#equation-delta" title="Link to this equation">¶</a></span>\[\delta = -\dfrac{1}{\rho}\left(\dfrac{\partial\rho}{\partial \xi}\right)_{p,T},\]</div>
<figure class="align-center" id="id1">
<a class="reference internal image-reference" href="_images/shell.png"><img alt="caption" src="_images/shell.png" style="width: 312.5px; height: 322.0px;" />
</a>
<figcaption>
<p><span class="caption-text">Sketch of the spherical shell model and its system of coordinate.</span><a class="headerlink" href="#id1" title="Link to this image">¶</a></p>
</figcaption>
</figure>
<section id="the-reference-state">
<h2>The reference state<a class="headerlink" href="#the-reference-state" title="Link to this heading">¶</a></h2>
<p>The convective flow and the related processes including magnetic field generation constitute
only small disturbances around a background or reference state. In the following we denote the background
state with a tilde and the disturbance we are interested in with a prime.
Formally we will solve equations in first order of a smallness parameters <span class="math notranslate nohighlight">\(\epsilon\)</span> which
quantified the ratio of convective disturbances to background state:</p>
<div class="math notranslate nohighlight" id="equation-epsilon">
<span class="eqno">(10)<a class="headerlink" href="#equation-epsilon" title="Link to this equation">¶</a></span>\[\epsilon \sim \dfrac{T'}{\tilde{T}} \sim \dfrac{p'}{\tilde{p}} \sim \dfrac{\rho'}{\tilde{\rho}} \sim ... \ll 1 .\]</div>
<p>The background state is hydrostatic, i.e. obeys the simple force balance</p>
<div class="math notranslate nohighlight" id="equation-hydrostatic">
<span class="eqno">(11)<a class="headerlink" href="#equation-hydrostatic" title="Link to this equation">¶</a></span>\[\nabla \tilde{p} = \tilde{\rho} \tilde{\vec{g}}.\]</div>
<p>Convective motions are supposed to be strong enough to provide homogeneous entropy
(and composition). The reference state is thus adiabatic and its gradients can be
expressed in terms of the pressure gradient <a class="reference internal" href="#equation-hydrostatic">(11)</a>:</p>
<div class="math notranslate nohighlight" id="equation-nablat">
<span class="eqno">(12)<a class="headerlink" href="#equation-nablat" title="Link to this equation">¶</a></span>\[\dfrac{\nabla\tilde{T}}{\tilde{T}} = \dfrac{1}{\tilde{T}} \left(\dfrac{\partial T}{\partial p}\right)_s
\nabla p = \dfrac{\alpha}{c_p} \tilde{\vec{g}} ,\]</div>
<div class="math notranslate nohighlight" id="equation-nablarho">
<span class="eqno">(13)<a class="headerlink" href="#equation-nablarho" title="Link to this equation">¶</a></span>\[\dfrac{\nabla\tilde{\rho}}{\tilde{\rho}} = \dfrac{1}{\tilde{\rho}} \left(\dfrac{\partial \rho}{\partial p}\right)_s
\nabla p = \beta \tilde{\rho} \tilde{\vec{g}} .\]</div>
<p>The reference state obviously dependence only on radius.
Dimensionless numbers quantifying the temperature and density gradients are called dissipation number <span class="math notranslate nohighlight">\(Di\)</span> and
compressibility parameter <span class="math notranslate nohighlight">\(Co\)</span> respectively:</p>
<div class="math notranslate nohighlight">
\[Di = \dfrac{\alpha d}{c_p} \tilde{g},\]</div>
<p>and</p>
<div class="math notranslate nohighlight">
\[Co = d \beta \tilde{\rho} \tilde{g}.\]</div>
<p>Here <span class="math notranslate nohighlight">\(d\)</span> is a typical length scale, for example the shell thickness of the problem.
The dissipation number is something like an inverse temperature scale hight while the
compressibility parameters is an inverse density scale hight.
The ratio of both numbers also helps to quantify the relative impact of temperature and
pressure on density variations:</p>
<div class="math notranslate nohighlight" id="equation-deltarho">
<span class="eqno">(14)<a class="headerlink" href="#equation-deltarho" title="Link to this equation">¶</a></span>\[\dfrac{\alpha \nabla T}{\beta \nabla \rho} \approx \alpha \tilde{T} \dfrac{Di}{Co}.\]</div>
<p>As an example we demonstrate how to derive the first order continuity equation here.
Using <span class="math notranslate nohighlight">\(\rho=\tilde{\rho}+\rho'\)</span> in <a class="reference internal" href="#equation-eqcontinuity">(2)</a> leads to</p>
<div class="math notranslate nohighlight">
\[\dfrac{\partial \tilde{\rho}}{\partial t} + \dfrac{\partial \rho'}{\partial t}
= - \vec{\nabla} \cdot \left( \tilde{\rho} \vec{u} \right)
- \vec{\nabla} \cdot \left( \rho' \vec{u} \right).\]</div>
<p>The zero order term vanishes since the background density is considered static (or actually changing very slowly
on very long time scales). The second term in the right hand side is obviously of second order.
The ratio of the remaining two terms can be estimated to also be of first order in <span class="math notranslate nohighlight">\(\epsilon\)</span>, meaning
that the time derivative of <span class="math notranslate nohighlight">\(\rho\)</span> is actually also of second order:</p>
<div class="math notranslate nohighlight">
\[\dfrac{\left[\partial \rho /\partial t\right]}{\left[\vec{\nabla} \cdot \rho \vec{u}\right]} \approx
\dfrac{\rho'}{\tilde{\rho}}\approx\epsilon\;\;.\]</div>
<p>Square brackets denote order of magnitude estimates here.
We have used the fact that the reference state is
static and assume time scale of changes are comparable (or slower) <span class="math notranslate nohighlight">\(\rho'\)</span> than the time
scales represented by <span class="math notranslate nohighlight">\(u\)</span> and that length scales
associated to the gradient operator are not too small.
We can then neglect local variations in <span class="math notranslate nohighlight">\(\rho'\)</span> which means that sound waves are filtered out.
This first order continuity equation thus simply reads:</p>
<div class="math notranslate nohighlight" id="equation-eqcontiuity1">
<span class="eqno">(15)<a class="headerlink" href="#equation-eqcontiuity1" title="Link to this equation">¶</a></span>\[\vec{\nabla} \cdot \left( \tilde{\rho} \vec{u} \right) = 0.\]</div>
<p>This defines the so-called anelastic approximation where sound waves are filtered out by
neglecting the local time derivative of density. This approximation is justified when
typical velocities are sufficiently smaller than the speed of sound.</p>
</section>
<section id="boussinesq-approximation">
<h2>Boussinesq approximation<a class="headerlink" href="#boussinesq-approximation" title="Link to this heading">¶</a></h2>
<p>For Earth the dissipation number and the compressibility parameter
are around <span class="math notranslate nohighlight">\(0.2\)</span> when temperature and density jump over the whole liquid core
are considered. This motivates the so called Boussinesq approximation where <span class="math notranslate nohighlight">\(Di\)</span> and
<span class="math notranslate nohighlight">\(Co\)</span> are assumed to vanish. The continuity equation <a class="reference internal" href="#equation-eqcontinuity">(2)</a> then simplifies further:</p>
<div class="math notranslate nohighlight">
\[\dfrac{1}{\tilde{\rho}} \vec{\nabla} \cdot \tilde{\rho} \vec{u} = \dfrac{\vec{u}}{\tilde{\rho}} \cdot \nabla \tilde{\rho}
+ \nabla\cdot\vec{u} \approx \nabla\cdot\vec{u} = 0.\]</div>
<p>When using typical number for Earth, <a class="reference internal" href="#equation-deltarho">(14)</a> becomes <span class="math notranslate nohighlight">\(0.05\)</span> so that pressure effects on density may be neglected.
The first order Navier-Stokes equation (after to zero order hydrostatic reference solution has been subtracted) then reads:</p>
<div class="math notranslate nohighlight" id="equation-eqnsb">
<span class="eqno">(16)<a class="headerlink" href="#equation-eqnsb" title="Link to this equation">¶</a></span>\[\tilde{\rho}\left(\dfrac{\partial \vec{u}}{\partial t}+ \vec{u}\cdot\vec{\nabla}\
\vec{u} \right) =-\vec{\nabla} p' -2\rho\vec{\Omega}\times\vec{u}
+ \alpha \tilde{g}_o T' \dfrac{\vec{r}}{r_o}
+ \delta \tilde{g}_o \xi' \dfrac{\vec{r}}{r_o}
+ \dfrac{1}{\mu_0}(\vec{\nabla}\times\vec{B})\times\vec{B}
+ \tilde{\rho} \nu \Delta \vec{u}.\]</div>
<p>Here <span class="math notranslate nohighlight">\(u\)</span> and <span class="math notranslate nohighlight">\(B\)</span> are understood as first order disturbances and
<span class="math notranslate nohighlight">\(p'\)</span> is the first order non-hydrostatic pressure and <span class="math notranslate nohighlight">\(T'\)</span> the
super-adiabatic temperature and <span class="math notranslate nohighlight">\(\xi\)</span> the super-adiabatic chemical
composition. Above we have adopted a simplification of the buoyancy term. In
the Boussinesq limit with vanishing <span class="math notranslate nohighlight">\(Co\)</span> and a small density difference
between a solid inner and a liquid outer core a linear gravity dependence
provides a reasonable approximation:</p>
<div class="math notranslate nohighlight">
\[\tilde{\vec{g}} = \tilde{g}_o \dfrac{\vec{r}}{r_o},\]</div>
<p>where we have chosen the gravity <span class="math notranslate nohighlight">\(\tilde{g}_o\)</span> at the outer boundary
radius <span class="math notranslate nohighlight">\(r_o\)</span> as reference.</p>
<p>The first order energy equation becomes</p>
<div class="math notranslate nohighlight" id="equation-eqentropyb">
<span class="eqno">(17)<a class="headerlink" href="#equation-eqentropyb" title="Link to this equation">¶</a></span>\[\tilde{\rho}\left(\dfrac{\partial T'}{\partial t}+\vec{u}\cdot \vec{\nabla} T' \right) =
\kappa \Delta T' + \epsilon,\]</div>
<p>where we have assumed a homogeneous <span class="math notranslate nohighlight">\(k\)</span> and neglected viscous and Ohmic
heating which can be shown to scale with <span class="math notranslate nohighlight">\(Di\)</span> as we discuss below.
Furthermore, we have used the simple relation</p>
<div class="math notranslate nohighlight">
\[\partial s \approx \dfrac{\tilde{\rho} c_p}{\tilde{T}} \partial T,\]</div>
<p>defined the thermal diffusivity</p>
<div class="math notranslate nohighlight">
\[\kappa = \dfrac{k}{\tilde{\rho} c_p},\]</div>
<p>and adjusted the definition of <span class="math notranslate nohighlight">\(\epsilon\)</span>. Finally the first order equation
for chemical composition becomes</p>
<div class="math notranslate nohighlight" id="equation-eqcompb">
<span class="eqno">(18)<a class="headerlink" href="#equation-eqcompb" title="Link to this equation">¶</a></span>\[\tilde{\rho}\left(\dfrac{\partial \xi'}{\partial t}+\vec{u}\cdot \vec{\nabla} \xi' \right) =
\kappa_\xi \Delta \xi' + \epsilon_\xi,\]</div>
<p>where we have assumed a homogeneous <span class="math notranslate nohighlight">\(k_\xi\)</span> and adjusted the definition of <span class="math notranslate nohighlight">\(\epsilon_\xi\)</span>.</p>
<p>MagIC solves a dimensionless form of the differential equations. Time is scaled
in units of the viscous diffusion time <span class="math notranslate nohighlight">\(d^2/\nu\)</span>, length in units of the
shell thickness <span class="math notranslate nohighlight">\(d\)</span>, temperature in units of the temperature drop
<span class="math notranslate nohighlight">\(\Delta T=T_o-T_i\)</span> over the shell, composition in units of the composition
drop <span class="math notranslate nohighlight">\(\Delta \xi = \xi_o-\xi_i\)</span> over the shell and magnetic field in units
<span class="math notranslate nohighlight">\((\mu\lambda\tilde{\rho}\Omega)^{1/2}\)</span>. Technically the transition to
the dimensionless form is achieved by the substitution</p>
<div class="math notranslate nohighlight">
\[r\rightarrow r\;d ,\quad t\rightarrow (d^2/\nu)\;t ,\quad
T\rightarrow \Delta T\;T ,\quad \xi\rightarrow \Delta\xi\;\xi ,\quad
B\rightarrow \left(\mu\lambda\tilde{\rho}\Omega\right)^{1/2}B\]</div>
<p>where <span class="math notranslate nohighlight">\(r\)</span> stands for any length. The next step then is to collect the
physical properties as a few possible characteristic dimensionless numbers.
Note that many different scalings and combinations of dimensionless numbers are
possible. For the Navier-Stokes equation in the Boussinesq limit MagIC uses the
form:</p>
<div class="math notranslate nohighlight" id="equation-eqnsboussinesq">
<span class="eqno">(19)<a class="headerlink" href="#equation-eqnsboussinesq" title="Link to this equation">¶</a></span>\[\left(\dfrac{\partial \vec{u}}{\partial t}+ \vec{u}\cdot\vec{\nabla}\
\vec{u} \right) =-\vec{\nabla} p' -\dfrac{2}{E}\vec{e_z}\times\vec{u}
+ \dfrac{Ra}{Pr} T' \dfrac{\vec{r}}{r_o}
+ \dfrac{Ra_\xi}{Sc} \xi' \dfrac{\vec{r}}{r_o}
+ \dfrac{1}{E Pm}(\vec{\nabla}\times\vec{B})\times\vec{B}
+ \Delta \vec{u},\]</div>
<p>where <span class="math notranslate nohighlight">\(\vec{e}_z\)</span> is the unit vector in the direction of the rotation
axis and the meaning of the pressure disturbance <span class="math notranslate nohighlight">\(p'\)</span> has been adjusted
to the new dimensionless equation form.</p>
</section>
<section id="anelastic-approximation">
<h2>Anelastic approximation<a class="headerlink" href="#anelastic-approximation" title="Link to this heading">¶</a></h2>
<p>The anelastic approximation adopts the simplified continuity <a class="reference internal" href="#equation-eqcontiuity1">(15)</a>.
The background state can be specified in different ways, for example by
providing profiles based on internal models and/or ab initio simulations.
We will assume a polytropic ideal gas in the following.</p>
<section id="analytical-solution-in-the-limit-of-an-ideal-gas">
<h3>Analytical solution in the limit of an ideal gas<a class="headerlink" href="#analytical-solution-in-the-limit-of-an-ideal-gas" title="Link to this heading">¶</a></h3>
<p>In the limit of an ideal gas which follows
<span class="math notranslate nohighlight">\(\tilde{p}=\tilde{\rho}\tilde{T}\)</span> and has <span class="math notranslate nohighlight">\(\alpha=1/\tilde{T}\)</span>, one
directly gets:</p>
<div class="math notranslate nohighlight">
\[\begin{split}\begin{aligned}
\dfrac{d \tilde{T}}{dr} & = -Di\,\tilde{g}(r), \\
\tilde{\rho} & = \tilde{T}^{1/(\gamma-1)},
\end{aligned}\end{split}\]</div>
<p>where <span class="math notranslate nohighlight">\(\gamma=c_p/c_v\)</span>. Note that we have moved to a dimensionless
formulations here, where all quantities have been normalized with their outer boundary values
and <span class="math notranslate nohighlight">\(Di\)</span> refers to the respective outer boundary value.
If we in addition make the assumption of a
centrally-condensed mass in the center of the spherical shell of radius
<span class="math notranslate nohighlight">\(r\in[r_i,r_o]\)</span>, i.e. <span class="math notranslate nohighlight">\(g\propto, 1/r^2\)</span>, this leads to</p>
<div class="math notranslate nohighlight">
\[\begin{split}\begin{aligned}
\tilde{T}(r) & =Di\frac{r_o^2}{r}+(1-Di\,r_o), \\
\tilde{\rho}(r) & = \tilde{T}^m, \\
Di & = \dfrac{r_i}{r_o}\left(\exp\dfrac{N_\rho}{m}-1\right),
\end{aligned}\end{split}\]</div>
<p>where <span class="math notranslate nohighlight">\(N_\rho=\ln(\tilde{\rho}_i/\tilde{\rho}_o)\)</span> is the number of density scale heights of the reference
state and <span class="math notranslate nohighlight">\(m=1/(\gamma-1)\)</span> is the polytropic index.</p>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<ul class="simple">
<li><p>The relationship between <span class="math notranslate nohighlight">\(N_\rho\)</span> and the dissipation number
<span class="math notranslate nohighlight">\(Di\)</span> directly depends on the gravity profile. The formula above
is only valid when <span class="math notranslate nohighlight">\(g\propto 1/r^2\)</span>.</p></li>
<li><p>In this formulation, when you change the polytropic index <span class="math notranslate nohighlight">\(m\)</span>, you
also change the nature of the fluid you’re modelling since you accordingly
modify <span class="math notranslate nohighlight">\(\gamma=c_p/c_v\)</span>.</p></li>
</ul>
</div>
</section>
<section id="anelastic-mhd-equations">
<h3>Anelastic MHD equations<a class="headerlink" href="#anelastic-mhd-equations" title="Link to this heading">¶</a></h3>
<p>In the most general formulation, all physical properties defining the
background state may vary with depth. Specific reference values must then be
chosen to provide a unique dimensionless formulations and we typically chose
outer boundary values here. The exception is the magnetic diffusivity where we
adopt the inner boundary value instead. The motivation is twofold: (i) it
allows an easier control of the possible continuous conductivity value in the
inner core; (ii) it is a more natural choice when modelling gas giants planets
which exhibit a strong electrical conductivity decay in the outer layer.</p>
<p>The time scale is then the viscous diffusion time <span class="math notranslate nohighlight">\(d^2/\nu_o\)</span> where
<span class="math notranslate nohighlight">\(\nu_o\)</span> is the kinematic viscosity at the outer boundary. Magnetic field
is expressed in units of <span class="math notranslate nohighlight">\((\rho_o\mu_0\lambda_i\Omega)^{1/2}\)</span>, where
<span class="math notranslate nohighlight">\(\rho_o\)</span> is the density at the outer boundary and <span class="math notranslate nohighlight">\(\lambda_i\)</span> is
the magnetic diffusivity at the <strong>inner</strong> boundary.</p>
<p>This leads to the following sets of dimensionless equations:</p>
<div class="math notranslate nohighlight" id="equation-eqnsnd">
<span class="eqno">(20)<a class="headerlink" href="#equation-eqnsnd" title="Link to this equation">¶</a></span>\[\left(\dfrac{\partial \vec{u}}{\partial t}+\vec{u}\cdot\vec{\nabla}\vec{u}\right)
= -\vec{\nabla}\left({\dfrac{p'}{\tilde{\rho}}}\right) - \dfrac{2}{E}\vec{e_z}\times\vec{u}
+ \dfrac{Ra}{Pr}\tilde{g} \,s'\,\vec{e_r}
+ \dfrac{Ra_\xi}{Sc}\tilde{g} \,\xi'\,\vec{e_r}
+\dfrac{1}{Pm\,E \,\tilde{\rho}}\left(\vec{\nabla}\times \vec{B}
\right)\times \vec{B}+ \dfrac{1}{\tilde{\rho}} \vec{\nabla}\cdot \mathsf{S},\]</div>
<div class="math notranslate nohighlight" id="equation-eqcontnd">
<span class="eqno">(21)<a class="headerlink" href="#equation-eqcontnd" title="Link to this equation">¶</a></span>\[\vec{\nabla}\cdot\tilde{\rho}\vec{u}=0,\]</div>
<div class="math notranslate nohighlight" id="equation-eqmagnd">
<span class="eqno">(22)<a class="headerlink" href="#equation-eqmagnd" title="Link to this equation">¶</a></span>\[\vec{\nabla}\cdot\vec{B}=0,\]</div>
<div class="math notranslate nohighlight" id="equation-eqcompnd">
<span class="eqno">(23)<a class="headerlink" href="#equation-eqcompnd" title="Link to this equation">¶</a></span>\[\tilde{\rho}\left(\dfrac{\partial \xi'}{\partial t} +
\vec{u}\cdot\vec{\nabla} \xi'\right) =
\dfrac{1}{Sc}\vec{\nabla}\cdot\left(\kappa_\xi(r)\tilde{\rho}\vec{\nabla} \xi'\right)\]</div>
<div class="math notranslate nohighlight" id="equation-eqindnd">
<span class="eqno">(24)<a class="headerlink" href="#equation-eqindnd" title="Link to this equation">¶</a></span>\[\dfrac{\partial \vec{B}}{\partial t} = \vec{\nabla} \times \left( \vec{u}\times\vec{B}\right)-\dfrac{1}{Pm}\vec{\nabla}\times\left(\lambda(r)\,\vec{\nabla}\times\vec{B}\right).\]</div>
<p>Here <span class="math notranslate nohighlight">\(\tilde{g}\)</span> and <span class="math notranslate nohighlight">\(\tilde{\rho}\)</span> are
the normalized radial gravity and density profiles that reach one at the outer boundary.</p>
</section>
<section id="entropy-equation-and-turbulent-diffusion">
<h3>Entropy equation and turbulent diffusion<a class="headerlink" href="#entropy-equation-and-turbulent-diffusion" title="Link to this heading">¶</a></h3>
<p>The entropy equation usually requires an additional assumption in most of the
existing anelastic approximations. Indeed, if one simply expands Eq.
<a class="reference internal" href="#equation-eqentropy">(4)</a> with the classical temperature diffusion an operator of the
form:</p>
<div class="math notranslate nohighlight">
\[\epsilon\,\vec{\nabla}\cdot \left( K \vec{\nabla} T'\right)+\vec{\nabla}\cdot \left( K \vec{\nabla} \tilde{T}\right),\]</div>
<p>will remain the right-hand side of the equation. At first glance, there seems
to be a <span class="math notranslate nohighlight">\(1/\epsilon\)</span> factor between the first term and the second one,
which would suggest to keep only the second term in this expansion. However,
for astrophysical objects which exhibit strong convective driving (and hence
large Rayleigh numbers), the diffusion of the adiabatic background is actually
very small and may be comparable or even smaller in magnitude than the <span class="math notranslate nohighlight">\(\epsilon\)</span>
terms representing the usual convective perturbations. For the Earth core for instance,
if one assumes that the typical temperature fluctuations are of the order of 1 mK and
the temperature contrast between the inner and outer core is of the order of 1000 K, then
<span class="math notranslate nohighlight">\(\epsilon \sim 10^{-6}\)</span>. The ratio of the two terms can thus be estimated as</p>
<div class="math notranslate nohighlight" id="equation-eqepsratio">
<span class="eqno">(25)<a class="headerlink" href="#equation-eqepsratio" title="Link to this equation">¶</a></span>\[\epsilon \dfrac{T'/\delta^2}{T/d^2},\]</div>
<p>where <span class="math notranslate nohighlight">\(d\)</span> is the thickness of the inner core and <span class="math notranslate nohighlight">\(\delta\)</span> is the typical thermal
boundary layer thickness. This ratio is exactly one when <span class="math notranslate nohighlight">\(\delta =1\text{ m}\)</span>, a
plausible value for the Earth inner core.</p>
<p>In numerical simulations however, the over-estimated diffusivities restrict the
computational capabilities to much lower Rayleigh numbers. As a consequence,
the actual boundary layers in a global DNS will be much thicker and the ratio
<a class="reference internal" href="#equation-eqepsratio">(25)</a> will be much smaller than unity. The second terms will thus
effectively acts as a radial-dependent heat source or sink that will drive or
hinder convection. This is one of the physical motivation to rather introduce a
<strong>turbulent diffusivity</strong> that will be approximated by</p>
<div class="math notranslate nohighlight">
\[\kappa \tilde{\rho}\tilde{T} \vec{\nabla} s,\]</div>
<p>where <span class="math notranslate nohighlight">\(\kappa\)</span> is the turbulent diffusivity. <strong>Entropy diffusion is assumed to dominate
over temperature diffusion in turbulent flows</strong>.</p>
<p>The choice of the entropy scale to non-dimensionalize Eq. <a class="reference internal" href="#equation-eqentropy">(4)</a> also
depends on the nature of the boundary conditions: it can be simply the entropy
contrast over the layer <span class="math notranslate nohighlight">\(\Delta s\)</span> when the entropy is held constant at
both boundaries, or <span class="math notranslate nohighlight">\(d\,(ds /dr)\)</span> when flux-based boundary conditions are
employed. We will restrict to the first option in the following, but keep in
mind that depending on your setup, the entropy reference scale (and thus the
Rayleigh number definition) might change.</p>
<div class="math notranslate nohighlight" id="equation-eqentropynd">
<span class="eqno">(26)<a class="headerlink" href="#equation-eqentropynd" title="Link to this equation">¶</a></span>\[\tilde{\rho}\tilde{T}\left(\dfrac{\partial s'}{\partial t} +
\vec{u}\cdot\vec{\nabla} s'\right) =
\dfrac{1}{Pr}\vec{\nabla}\cdot\left(\kappa(r)\tilde{\rho}\tilde{T}\vec{\nabla} s'\right) +
\dfrac{Pr\,Di}{Ra}\Phi_\nu +
\dfrac{Pr\,Di}{Pm^2\,E\,Ra}\lambda(r)\left(\vec{\nabla}
\times\vec{B}\right)^2,\]</div>
<p>A comparison with <a class="reference internal" href="#equation-eqnsnd">(20)</a> reveals meaning of the different non-dimensional
numbers that scale viscous and Ohmic heating. The fraction <span class="math notranslate nohighlight">\(Pr/Ra\)</span> simply
expresses the ratio of entropy and flow in the Navier-Stokes equation, while
the additional factor <span class="math notranslate nohighlight">\(1/E Pm\)</span> reflects the scale difference of magnetic
field and flow. Then remaining dissipation number <span class="math notranslate nohighlight">\(Di\)</span> then expresses
the relative importance of viscous and Ohmic heating compared to buoyancy and
Lorentz force in the Navier-Stokes equation. For small <span class="math notranslate nohighlight">\(Di\)</span> both heating
terms can be neglected compared to entropy changes due to advection, an limit
that is used in the Boussinesq approximation.</p>
</section>
</section>
<section id="dimensionless-control-parameters">
<h2>Dimensionless control parameters<a class="headerlink" href="#dimensionless-control-parameters" title="Link to this heading">¶</a></h2>
<p>The equations <a class="reference internal" href="#equation-eqnsnd">(20)</a>-<a class="reference internal" href="#equation-eqentropynd">(26)</a> are governed by four dimensionless numbers: the
Ekman number</p>
<div class="math notranslate nohighlight" id="equation-eqekman">
<span class="eqno">(27)<a class="headerlink" href="#equation-eqekman" title="Link to this equation">¶</a></span>\[E = \frac{\nu}{\Omega d^2},\]</div>
<p>the thermal Rayleigh number</p>
<div class="math notranslate nohighlight" id="equation-eqrayleigh">
<span class="eqno">(28)<a class="headerlink" href="#equation-eqrayleigh" title="Link to this equation">¶</a></span>\[Ra = \frac{\alpha_o g_o T_o d^3 \Delta s}{c_p \kappa_o \nu_o},\]</div>
<p>the compositional Rayleigh number</p>
<div class="math notranslate nohighlight" id="equation-eqrayleighxi">
<span class="eqno">(29)<a class="headerlink" href="#equation-eqrayleighxi" title="Link to this equation">¶</a></span>\[Ra_\xi = \frac{\delta_o g_o d^3 \Delta \xi}{\kappa_\xi \nu_o},\]</div>
<p>the Prandtl number</p>
<div class="math notranslate nohighlight" id="equation-eqprandtl">
<span class="eqno">(30)<a class="headerlink" href="#equation-eqprandtl" title="Link to this equation">¶</a></span>\[Pr = \frac{\nu_o}{\kappa_o},\]</div>
<p>the Schmidt number</p>
<div class="math notranslate nohighlight" id="equation-eqschmidt">
<span class="eqno">(31)<a class="headerlink" href="#equation-eqschmidt" title="Link to this equation">¶</a></span>\[Sc = \frac{\nu_o}{\kappa_\xi},\]</div>
<p>and the magnetic Prandtl number</p>
<div class="math notranslate nohighlight" id="equation-eqmaprandtl">
<span class="eqno">(32)<a class="headerlink" href="#equation-eqmaprandtl" title="Link to this equation">¶</a></span>\[Pm = \frac{\nu_o}{\lambda_i}.\]</div>
<p>In addition to these four numbers, the reference state is controlled by the geometry of
the spherical shell given by its radius ratio</p>
<div class="math notranslate nohighlight" id="equation-eqradratio">
<span class="eqno">(33)<a class="headerlink" href="#equation-eqradratio" title="Link to this equation">¶</a></span>\[\eta = \frac{r_i}{r_o},\]</div>
<p>and the background density and temperature profiles, either controlled by <span class="math notranslate nohighlight">\(Di\)</span> or
by <span class="math notranslate nohighlight">\(N_\rho\)</span> and <span class="math notranslate nohighlight">\(m\)</span>.</p>
<p>In the Boussinesq approximation all physical properties are assumed to
be homogeneous and we can drop the sub-indices <span class="math notranslate nohighlight">\(o\)</span> and <span class="math notranslate nohighlight">\(i\)</span>
except for gravity.
Moreover, the Rayleigh number can be expressed in terms of the temperature
jump across the shell:</p>
<div class="math notranslate nohighlight" id="equation-eqrayleighboussinesq">
<span class="eqno">(34)<a class="headerlink" href="#equation-eqrayleighboussinesq" title="Link to this equation">¶</a></span>\[Ra = \frac{\alpha g_o d^3 \Delta T}{\kappa \nu}.\]</div>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p>In MagIC, those control parameters can be adjusted in the
<a class="reference internal" href="inputNamelists/physNamelist.html#secphysnml"><span class="std std-ref">&phys_param</span></a> section of the input namelist.</p>
</div>
<p>Variants of the non-dimensional equations and control parameters result from
different choices for the fundamental scales. For the length scale often
<span class="math notranslate nohighlight">\(r_o\)</span> is chosen instead of <span class="math notranslate nohighlight">\(d\)</span>. Other natural scales for time are the
magnetic or the thermal diffusion time, or the rotation period.
There are also different options for scaling the magnetic field strength.
The prefactor of two, which is retained in the
Coriolis term in <a class="reference internal" href="#equation-eqnsnd">(20)</a>, is often incorporated into the definition of the
Ekman number.</p>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p>Those references timescales and length scales can be adjusted by
several input parameters in the <a class="reference internal" href="inputNamelists/controlNamelist.html#seccontrolnml"><span class="std std-ref">&control</span></a> section
of the input namelist.</p>
</div>
<section id="usual-diagnostic-quantities">
<h3>Usual diagnostic quantities<a class="headerlink" href="#usual-diagnostic-quantities" title="Link to this heading">¶</a></h3>
<p>Characteristic properties of the solution are usually expressed in terms
of non-dimensional diagnostic parameters.
In the context of the geodynamo for instance, the two
most important ones are the magnetic Reynolds number <span class="math notranslate nohighlight">\(Rm\)</span> and
the Elsasser number <span class="math notranslate nohighlight">\(\Lambda\)</span>. Usually the rms-values of the velocity
<span class="math notranslate nohighlight">\(u_{rms}\)</span> and of the magnetic field <span class="math notranslate nohighlight">\(B_{rms}\)</span> inside the spherical shell
are taken as characteristic values. The magnetic Reynolds number</p>
<div class="math notranslate nohighlight">
\[Rm = \frac{u_{rms}d}{\lambda_i}\]</div>
<p>can be considered as a measure for the flow velocity and describes
the ratio of advection of the magnetic field to magnetic diffusion.
Other characteristic non-dimensional numbers related to the flow velocity are
the (hydrodynamic) Reynolds number</p>
<div class="math notranslate nohighlight">
\[Re = \frac{u_{rms} d}{\nu_o},\]</div>
<p>which measures the ratio of inertial forces to viscous forces,
and the Rossby number</p>
<div class="math notranslate nohighlight">
\[Ro = \frac{u_{rms}}{\Omega d} ,\]</div>
<p>a measure for the ratio of inertial to Coriolis forces.</p>
<div class="math notranslate nohighlight">
\[\Lambda = \frac{B_{rms}^2}{\mu_0\lambda_i\rho_o\Omega}\]</div>
<p>measures the ratio of Lorentz to Coriolis forces and is
equivalent to the square of the non-dimensional magnetic field strength
in the scaling chosen here.</p>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p>The time-evolution of these diagnostic quantities are stored in
the <a class="reference internal" href="outputFiles/outTimeSeriesStd.html#secparfile"><span class="std std-ref">par.TAG</span></a> file produced during the run of MagIC.</p>
</div>
</section>
</section>
<section id="boundary-conditions-and-treatment-of-inner-core">
<h2>Boundary conditions and treatment of inner core<a class="headerlink" href="#boundary-conditions-and-treatment-of-inner-core" title="Link to this heading">¶</a></h2>
<section id="mechanical-conditions">
<h3>Mechanical conditions<a class="headerlink" href="#mechanical-conditions" title="Link to this heading">¶</a></h3>
<p>In its simplest form, when modelling the geodynamo, the fluid shell is treated
as a container with rigid, impenetrable, and co-rotating walls. This implies
that within the rotating frame of reference all velocity components vanish at
<span class="math notranslate nohighlight">\(r_o\)</span> and <span class="math notranslate nohighlight">\(r_i\)</span>. In case of modelling the free surface of a gas
giant planets or a star, it is preferable to rather replace the condition of
zero horizontal velocity by one of vanishing viscous shear stresses (the
so-called free-slip condition).</p>
<p>Furthermore, even in case of modelling the liquid iron core of a terrestrial
planet, there is no a priori reason why the inner core should necessarily
co-rotate with the mantle. Some models for instance allow for differential
rotation of the inner core and mantle with respect to the reference frame. The
change of rotation rate is determined from the net torque. Viscous,
electromagnetic, and torques due to gravitational coupling between density
heterogeneities in the mantle and in the inner core contribute.</p>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p>The mechanical boundary conditions can be adjusted with the
parameters <a class="reference internal" href="inputNamelists/physNamelist.html#secmechanicalbcs"><span class="std std-ref">ktopv</span></a> and
<a class="reference internal" href="inputNamelists/physNamelist.html#secmechanicalbcs"><span class="std std-ref">kbotv</span></a> in the <a class="reference internal" href="inputNamelists/physNamelist.html#secphysnml"><span class="std std-ref">&phys_param</span></a>
section of the input namelist.</p>
</div>
</section>
<section id="magnetic-boundary-conditions-and-inner-core-conductivity">
<h3>Magnetic boundary conditions and inner core conductivity<a class="headerlink" href="#magnetic-boundary-conditions-and-inner-core-conductivity" title="Link to this heading">¶</a></h3>
<p>When assuming that the fluid shell is surrounded by electrically insulating regions
(inner core and external part),
the magnetic field inside the fluid shell matches continuously
to a potential field in both the exterior and the interior regions. Alternative
magnetic boundary conditions (like cancellation of the horizontal component of the field
) are also possible.</p>
<p>Depending on the physical problem you want to model, treating the inner core as an
insulator is not realistic either, and it might instead be more appropriate to
assume that it has the same electrical conductivity as
the fluid shell. In this case, an equation equivalent to <a class="reference internal" href="#equation-eqindnd">(24)</a> must
be solved for the inner core, where the velocity field simply
describes the solid body rotation of the inner core with respect to the reference frame.
At the inner core boundary a continuity condition for the magnetic field and the
horizontal component of the electrical field apply.</p>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p>The magnetic boundary conditions can be adjusted with the parameters
<a class="reference internal" href="inputNamelists/physNamelist.html#secmagneticbcs"><span class="std std-ref">ktopb</span></a> and <a class="reference internal" href="inputNamelists/physNamelist.html#secmagneticbcs"><span class="std std-ref">kbotb</span></a>
in the <a class="reference internal" href="inputNamelists/physNamelist.html#secphysnml"><span class="std std-ref">&phys_param</span></a> section of the input namelist.</p>
</div>
</section>
<section id="thermal-boundary-conditions-and-distribution-of-buoyancy-sources">
<h3>Thermal boundary conditions and distribution of buoyancy sources<a class="headerlink" href="#thermal-boundary-conditions-and-distribution-of-buoyancy-sources" title="Link to this heading">¶</a></h3>
<p>In many dynamo models, convection is simply driven by an imposed fixed
super-adiabatic entropy contrast between the inner and outer boundaries. This
approximation is however not necessarily the best choice, since for instance,
in the present Earth, convection is thought to be driven by a combination of
thermal and compositional buoyancy. Sources of heat are the release of latent
heat of inner core solidification and the secular cooling of the outer and
inner core, which can effectively be treated like a heat source. The heat loss
from the core is controlled by the convecting mantle, which effectively imposes
a condition of fixed heat flux at the core-mantle boundary on the dynamo. The
heat flux is in that case spatially and temporally variable.</p>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p>The thermal boundary conditions can be adjusted with the parameters
<a class="reference internal" href="inputNamelists/physNamelist.html#secthermalbcs"><span class="std std-ref">ktops</span></a> and <a class="reference internal" href="inputNamelists/physNamelist.html#secthermalbcs"><span class="std std-ref">kbots</span></a>
in the <a class="reference internal" href="inputNamelists/physNamelist.html#secphysnml"><span class="std std-ref">&phys_param</span></a> section of the input namelist.</p>
</div>
</section>
<section id="chemical-composition-boundary-conditions">
<h3>Chemical composition boundary conditions<a class="headerlink" href="#chemical-composition-boundary-conditions" title="Link to this heading">¶</a></h3>
<p>They are treated in a very similar manner as the thermal boundary conditions</p>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p>The boundary conditions for composition can be adjusted with the parameters
<a class="reference internal" href="inputNamelists/physNamelist.html#seccompbcs"><span class="std std-ref">ktopxi</span></a> and <a class="reference internal" href="inputNamelists/physNamelist.html#seccompbcs"><span class="std std-ref">kbotxi</span></a>
in the <a class="reference internal" href="inputNamelists/physNamelist.html#secphysnml"><span class="std std-ref">&phys_param</span></a> section of the input namelist.</p>
</div>
</section>
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<h3><a href="contents.html">Table of Contents</a></h3>
<ul>
<li><a class="reference internal" href="#">Formulation of the (magneto)-hydrodynamics problem</a><ul>
<li><a class="reference internal" href="#the-reference-state">The reference state</a></li>
<li><a class="reference internal" href="#boussinesq-approximation">Boussinesq approximation</a></li>
<li><a class="reference internal" href="#anelastic-approximation">Anelastic approximation</a><ul>
<li><a class="reference internal" href="#analytical-solution-in-the-limit-of-an-ideal-gas">Analytical solution in the limit of an ideal gas</a></li>
<li><a class="reference internal" href="#anelastic-mhd-equations">Anelastic MHD equations</a></li>
<li><a class="reference internal" href="#entropy-equation-and-turbulent-diffusion">Entropy equation and turbulent diffusion</a></li>
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</li>
<li><a class="reference internal" href="#dimensionless-control-parameters">Dimensionless control parameters</a><ul>
<li><a class="reference internal" href="#usual-diagnostic-quantities">Usual diagnostic quantities</a></li>
</ul>
</li>
<li><a class="reference internal" href="#boundary-conditions-and-treatment-of-inner-core">Boundary conditions and treatment of inner core</a><ul>
<li><a class="reference internal" href="#mechanical-conditions">Mechanical conditions</a></li>
<li><a class="reference internal" href="#magnetic-boundary-conditions-and-inner-core-conductivity">Magnetic boundary conditions and inner core conductivity</a></li>
<li><a class="reference internal" href="#thermal-boundary-conditions-and-distribution-of-buoyancy-sources">Thermal boundary conditions and distribution of buoyancy sources</a></li>
<li><a class="reference internal" href="#chemical-composition-boundary-conditions">Chemical composition boundary conditions</a></li>
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