96 research outputs found
Thermal convection in Earth's inner core with phase change at its boundary
Inner core translation, with solidification on one hemisphere and melting on
the other, provides a promising basis for understanding the hemispherical
dichotomy of the inner core, as well as the anomalous stable layer observed at
the base of the outer core - the F-layer - which might be sustained by
continuous melting of inner core material. In this paper, we study in details
the dynamics of inner core thermal convection when dynamically induced melting
and freezing of the inner core boundary (ICB) are taken into account. If the
inner core is unstably stratified, linear stability analysis and numerical
simulations consistently show that the translation mode dominates only if the
viscosity is large enough, with a critical viscosity value, of order Pas, depending on the ability of outer core convection to supply or
remove the latent heat of melting or solidification. If is smaller, the
dynamical effect of melting and freezing is small. Convection takes a more
classical form, with a one-cell axisymmetric mode at the onset and chaotic
plume convection at large Rayleigh number. [...] Thermal convection requires
that a superadiabatic temperature profile is maintained in the inner core,
which depends on a competition between extraction of the inner core internal
heat by conduction and cooling at the ICB. Inner core thermal convection
appears very likely with the low thermal conductivity value proposed by Stacey
& Davis (2007), but nearly impossible with the much higher thermal conductivity
recently put forward. We argue however that the formation of an iron-rich layer
above the ICB may have a positive feedback on inner core convection: it implies
that the inner core crystallized from an increasingly iron-rich liquid,
resulting in an unstable compositional stratification which could drive inner
core convection, perhaps even if the inner core is subadiabatic.Comment: 25 pages, 12 figure
Bound of dissipation on a plane Couette dynamo
Variational turbulence is among the few approaches providing rigorous results
in turbulence. In addition, it addresses a question of direct practical
interest, namely the rate of energy dissipation. Unfortunately, only an upper
bound is obtained as a larger functional space than the space of solutions to
the Navier-Stokes equations is searched. Yet, in general, this upper bound is
in good agreement with experimental results in terms of order of magnitude and
power law of the imposed Reynolds number. In this paper, the variational
approach to turbulence is extended to the case of dynamo action and an upper
bound is obtained for the global dissipation rate (viscous and Ohmic). A simple
plane Couette flow is investigated. For low magnetic Prandtl number
fluids, the upper bound of energy dissipation is that of classical turbulence
(i.e. proportional to the cubic power of the shear velocity) for magnetic
Reynolds numbers below and follows a steeper evolution for magnetic
Reynolds numbers above (i.e. proportional to the shear velocity to
the power four) in the case of electrically insulating walls. However, the
effect of wall conductance is crucial : for a given value of wall conductance,
there is a value for the magnetic Reynolds number above which energy
dissipation cannot be bounded. This limiting magnetic Reynolds number is
inversely proportional to the square root of the conductance of the wall.
Implications in terms of energy dissipation in experimental and natural dynamos
are discussed.Comment: In this new version, amistake (in equation 23 of the first version)
is correcte
Remarks on compressible convection in Super-Earths
The radial density of planets increases with depth due to compressibility,
leading to impacts on their convective dynamics. To account for these effects,
including the presence of a quasi-adiabatic temperature profile and entropy
sources due to dissipation, the compressibility is expressed through a
dissipation number, , proportional to the planet's radius and
gravity. In Earth's mantle, compressibility effects are moderate, but in large
rocky or liquid exoplanets (Super-Earths), the dissipation number can become
very large. This paper explores the properties of compressible convection when
the dissipation number is significant. We start by selecting a simple Murnaghan
equation of state that embodies the fundamental properties of condensed matter
at planetary conditions. Next, we analyze the characteristics of adiabatic
profiles and demonstrate that the ratio between the bottom and top adiabatic
temperatures is relatively small and probably less than 2. We examine the
marginal stability of compressible mantles and reveal that they can undergo
convection with either positive or negative superadiabatic Rayleigh numbers.
Lastly, we delve into simulations of convection performed using the exact
equations of mechanics, neglecting inertia (infinite Prandtl number case), and
examine their consequences for Super-Earths dynamics.Comment: 31 pages, 15 figure
Axisymmetric dynamo action produced by differential rotation, with anisotropic electrical conductivity and anisotropic magnetic permeability
The effect on dynamo action of an anisotropic electrical conductivity
conjugated to an anisotropic magnetic permeability is considered. Not only is
the dynamo fully axisymmetric, but it requires only a simple differential
rotation, which twice challenges the well-established dynamo theory. Stability
analysis is conducted entirely analytically, leading to an explicit expression
of the dynamo threshold. The results show a competition between the anisotropy
of electrical conductivity and that of magnetic permeability, the dynamo effect
becoming impossible if the two anisotropies are identical. For isotropic
electrical conductivity, Cowling's neutral point argument does imply the
absence of an azimuthal component of current density, but does not prevent the
dynamo effect as long as the magnetic permeability is anisotropic.Comment: 19 pages, 6 figure
A case for variational geomagnetic data assimilation: insights from a one-dimensional, nonlinear, and sparsely observed MHD system
Secular variations of the geomagnetic field have been measured with a continuously improving accuracy during the last few hundred years, culminating nowadays with satellite data. It is however well known that the dynamics of the magnetic field is linked to that of the velocity field in the core and any attempt to model secular variations will involve a coupled dynamical system for magnetic field and core velocity. Unfortunately, there is no direct observation of the velocity. Independently of the exact nature of the above-mentioned coupled system -- some version being currently under construction -- the question is debated in this paper whether good knowledge of the magnetic field can be translated into good knowledge of core dynamics. Furthermore, what will be the impact of the most recent and precise geomagnetic data on our knowledge of the geomagnetic field of the past and future? These questions are cast into the language of variational data assimilation, while the dynamical system considered in this paper consists in a set of two oversimplified one-dimensional equations for magnetic and velocity fields. This toy model retains important features inherited from the induction and Navier-Stokes equations: non-linear magnetic and momentum terms are present and its linear response to small disturbances contains Alfvén waves. It is concluded that variational data assimilation is indeed appropriate in principle, even though the velocity field remains hidden at all times; it allows us to recover the entire evolution of both fields from partial and irregularly distributed information on the magnetic field. This work constitutes a first step on the way toward the reassimilation of historical geomagnetic data and geomagnetic forecast
On the stability of the Hartmann layer
In this paper we are concerned with the theoretical stability of the laminar Hartmann layer, which forms at the boundary of any electrically conducting fluid flow under a steady magnetic field at high Hartmann number. We perform both linear and energetic stability analyses to investigate the stability of the Hartmann layer to both infinitesimal and finite perturbations. We find that there is more than three orders of magnitude between the critical Reynolds numbers from these two analyses. Our interest is motivated by experimental results on the laminar–turbulent transition of ducted magnetohydrodynamics flows. Importantly, all existing experiments have considered the laminarization of a turbulent flow, rather than transition to turbulence. The fact that experiments have considered laminarization, rather than transition, implies that the threshold value of the Reynolds number for stability of the Hartmann layer to finite-amplitude, rather than infinitesimal, disturbances is in better agreement with the experimental threshold values. In fact, the critical Reynolds number for linear instability of the Hartmann layer is more than two orders of magnitude larger than experimentally observed threshold values. It seems that this large discrepancy has led to the belief that stability or instability of the Hartmann layer has no bearing on whether the flow is laminar or turbulent. In this paper, we give support to Lock’s hypothesis [Proc. R. Soc. London, Ser. A 233, 105 (1955)] that “transition” is due to the stability characteristics of the Hartmann layer with respect to large-amplitude disturbances
A model for the turbulent Hartmann layer
Here we study the Hartmann layer, which forms at the boundary of any electrically-conducting fluid
flow under a steady magnetic field at high Hartmann number provided the magnetic field is not
parallel to the wall. The Hartmann layer has a well-known form when laminar. In this paper we
develop a model for the turbulent Hartmann layer based on Prandtl’s mixing-length model without
adding arbitrary parameters, other than those already included in the log-law. We find an exact
expression for the displacement thickness of the turbulent Hartmann layer @also given by Tennekes,
Phys. Fluids 9, 1876 ~1966!#, which supports our assertion that a fully-developed turbulent
Hartmann layer of finite extent exists. Leading from this expression, we show that the interaction
parameter is small compared with unity and that therefore the Lorentz force is negligible compared
with inertia. Hence, we suggest that the turbulence present in the Hartmann layer is of classical type
and not affected by the imposed magnetic field, so justifying use of a Prandtl model. A major result
is a simple implicit relationship between the Reynolds number and the friction coefficient for the
turbulent Hartmann layer in the limit of large Reynolds number. By considering the distance over
which the stress decays, we find a condition for the two opposite Hartmann layers in duct flows to
be isolated (nonoverlapping)
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