Formulation of the (magneto)-hydrodynamics problem¶
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
The conservation of momentum is formulated by the Navier-Stokes equation
where
Convection is driven by buoyancy forces acting on variations in density
In addition an equation of state is required to formulate the thermodynamic density changes. For example the relation
describes density variations caused by variations in temperature
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
where
Note that we use here the summation convention over the indices
The induction equation is obtained from the Maxwell equations (ignoring displacement current) and Ohm’s law (neglecting Hall effect):
When the magnetic diffusivity
The physical properties determining above equations are rotation rate
the compressibility at constant temperature
and an equivalent parameter

Sketch of the spherical shell model and its system of coordinate.¶
The reference state¶
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
The background state is hydrostatic, i.e. obeys the simple force balance
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 (11):
The reference state obviously dependence only on radius.
Dimensionless numbers quantifying the temperature and density gradients are called dissipation number
and
Here
As an example we demonstrate how to derive the first order continuity equation here.
Using
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
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)
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.
Boussinesq approximation¶
For Earth the dissipation number and the compressibility parameter
are around
When using typical number for Earth, (14) becomes
Here
where we have chosen the gravity
The first order energy equation becomes
where we have assumed a homogeneous
defined the thermal diffusivity
and adjusted the definition of
where we have assumed a homogeneous
MagIC solves a dimensionless form of the differential equations. Time is scaled
in units of the viscous diffusion time
where
where
Anelastic approximation¶
The anelastic approximation adopts the simplified continuity (15). 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.
Analytical solution in the limit of an ideal gas¶
In the limit of an ideal gas which follows
where
where
Warning
The relationship between
and the dissipation number directly depends on the gravity profile. The formula above is only valid when .In this formulation, when you change the polytropic index
, you also change the nature of the fluid you’re modelling since you accordingly modify .
Anelastic MHD equations¶
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.
The time scale is then the viscous diffusion time
This leads to the following sets of dimensionless equations:
Here
Entropy equation and turbulent diffusion¶
The entropy equation usually requires an additional assumption in most of the existing anelastic approximations. Indeed, if one simply expands Eq. (4) with the classical temperature diffusion an operator of the form:
will remain the right-hand side of the equation. At first glance, there seems
to be a
where
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 (25) 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 turbulent diffusivity that will be approximated by
where
The choice of the entropy scale to non-dimensionalize Eq. (4) also
depends on the nature of the boundary conditions: it can be simply the entropy
contrast over the layer
A comparison with (20) reveals meaning of the different non-dimensional
numbers that scale viscous and Ohmic heating. The fraction
Dimensionless control parameters¶
The equations (20)-(26) are governed by four dimensionless numbers: the Ekman number
the thermal Rayleigh number
the compositional Rayleigh number
the Prandtl number
the Schmidt number
and the magnetic Prandtl number
In addition to these four numbers, the reference state is controlled by the geometry of the spherical shell given by its radius ratio
and the background density and temperature profiles, either controlled by
In the Boussinesq approximation all physical properties are assumed to
be homogeneous and we can drop the sub-indices
See also
In MagIC, those control parameters can be adjusted in the &phys_param section of the input namelist.
Variants of the non-dimensional equations and control parameters result from
different choices for the fundamental scales. For the length scale often
See also
Those references timescales and length scales can be adjusted by several input parameters in the &control section of the input namelist.
Usual diagnostic quantities¶
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
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
which measures the ratio of inertial forces to viscous forces, and the Rossby number
a measure for the ratio of inertial to Coriolis forces.
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.
See also
The time-evolution of these diagnostic quantities are stored in the par.TAG file produced during the run of MagIC.
Boundary conditions and treatment of inner core¶
Mechanical conditions¶
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
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.
See also
The mechanical boundary conditions can be adjusted with the parameters ktopv and kbotv in the &phys_param section of the input namelist.
Magnetic boundary conditions and inner core conductivity¶
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.
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 (24) 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.
See also
The magnetic boundary conditions can be adjusted with the parameters ktopb and kbotb in the &phys_param section of the input namelist.
Thermal boundary conditions and distribution of buoyancy sources¶
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.
See also
The thermal boundary conditions can be adjusted with the parameters ktops and kbots in the &phys_param section of the input namelist.
Chemical composition boundary conditions¶
They are treated in a very similar manner as the thermal boundary conditions
See also
The boundary conditions for composition can be adjusted with the parameters ktopxi and kbotxi in the &phys_param section of the input namelist.