Yue-Kin Tsang

The ratio of the magnetic power spectrum and the secular variation spectrum measured at the Earth’s surface provides a time scale τ_sv(l) as a function of spherical harmonic degree l. τ_sv is often assumed to be representative of time scales related to the dynamo inside the outer core and its scaling with l is debated. To assess the validity of this surmise and to study the time variation of the geomagnetic field |$\dot{B}$| inside the outer core, we introduce a magnetic time-scale spectrum τ(l, r) that is valid for all radius r above the inner core and reduces to the usual τ_sv at and above the core–mantle boundary (CMB). We study τ in a numerical geodynamo model. At the CMB, we find that τ ∼ l⁻¹ is valid at both the large and small scales, in agreement with previous numerical studies on τ_sv. Just below the CMB, the scaling undergo a sharp transition at small l. Consequently, in the interior of the outer core, τ exhibits different scaling at the large and small scales, specifically, the scaling of τ becomes shallower than l⁻¹ at small l. We find that this transition at the large scales stems from the fact that the horizontal components of the magnetic field evolve faster than the radial component in the interior. In contrast, the magnetic field at the CMB must match onto a potential field, hence the dynamics of the radial and horizontal magnetic fields are tied together. The upshot is τ_sv becomes unreliable in estimating time scales inside the outer core. Another question concerning τ is whether an argument based on the frozen-flux hypothesis can be used to explain its scaling. To investigate this, we analyse the induction equation in the spectral space. We find that away from both boundaries, the magnetic diffusion term is negligible in the power spectrum of |$\dot{B}$|⁠. However, |$\dot{B}$| is controlled by the radial derivative in the induction term, thus invalidating the frozen-flux argument. Near the CMB, magnetic diffusion starts to affect |$\dot{B}$| rendering the frozen-flux hypothesis inapplicable. We also examine the effects of different velocity boundary conditions and find that the above results apply for both no-slip and stress-free conditions at the CMB.

Recent satellite missions provide accurate measurement of the time derivative of the Gaussian coefficients from which the secular variation spectrum can be calculated. The ratio of the magnetic energy spectrum to the secular variation spectrum gives a typical scale τ for the temporal variation of the geomagnetic field as a function of the spherical harmonics degrees l. There is much interest in the scaling of τ with l: τ ~ l β. Numerical simulations and the frozen flux hypothesis suggest the simple relation τ ~ l -1 while observational studies give a diverse range of value for β. A question here is whether the frozen flux hypothesis is applicable. It is plausible that magnetic diffusion can be neglected inside the outer core. However, the situation in a boundary layer under the core-mantle boundary (CMB) is less clear. A related question is whether τ observed at the Earth's surface is relevant to what is happening in the interior of the outer core as the form of the magnetic field above the CMB is constrained by the boundary conditions at the CMB. Here we use a numerical dynamo model to investigate these questions. We extend the definition of τ to the inside of the outer core. We find that in our simulations the exponent β undergoes a sharp transition just beneath the CMB, magnetic diffusion plays a role in the scaling of τ above the CMB and the frozen flux hypothesis is not applicable here.

Jupiter's magnetic field is generated by the convection of liquid metallic hydrogen in its interior. The transition from molecular hydrogen to metallic hydrogen as temperature and pressure increase is believed to be a smooth one. As a result, the electrical conductivity in Jupiter varies continuously from being negligible at the surface to a large value in the deeper region. Thus, unlike the Earth where the upper boundary of the dynamo---the dynamo radius---is definitively located at the core-mantle boundary, it is not clear at what depth dynamo action becomes significant in Jupiter. In this paper, using a numerical model of the Jovian dynamo, we examine the magnetic energy spectrum at different depth and identify a dynamo radius below which (and away from the deep inner core) the shape of the magnetic energy spectrum becomes invariant. We find that this shift in the behaviour of the magnetic energy spectrum signifies a change in the dynamics of the system as electric current becomes important. Traditionally, a characteristic radius derived from the Lowes--Mauersberger spectrum---the Lowes radius---gives a good estimate to the Earth's core-mantle boundary. We argue that in our model, the Lowes radius provides a lower bound to the dynamo radius. We also compare the Lowes--Mauersberger spectrum in our model to that obtained from recent Juno observations. The Lowes radius derived from the Juno data is significantly lower than that obtained from our models. The existence of a stably stratified region in the neighbourhood of the transition zone might provide an explanation of this result.

In this paper we describe the construction of an efficient probabilistic parameterization that could be used in a coarse-resolution numerical model in which the variation of moisture is not properly resolved. An Eulerian model using a coarse-grained field on a grid cannot properly resolve regions of saturation---in which condensation occurs---that are smaller than the grid boxes. Thus, in the absence of a parameterization scheme, either the grid box must become saturated or condensation will be underestimated. On the other hand, in a stochastic Lagrangian model of moisture transport, trajectories of parcels tagged with humidity variables are tracked and small-scale moisture variability can be retained; however, explicitly implementing such a scheme in a global model would be computationally prohibitive. One way to introduce subgrid-scale saturation into an Eulerian model is to assume the humidity within a grid box has a probability distribution. To close the problem, this distribution is conventionally determined by relating the required subgrid-scale properties of the flow to the grid-scale properties using a turbulence closure. Here, instead, we determine an assumed probability distribution by using the statistical moments from a stochastic Lagrangian version of the system. The stochastic system is governed by a Fokker--Planck equation and we use that, rather than explicitly following the moisture parcels, to determine the parameters of the assumed distribution. We are thus able to parameterize subgrid-scale condensation in an Eulerian model in a computationally efficient and theoretically well-founded way. In two idealized advection--condensation problems we show that a coarse Eulerian model with the subgrid parameterization is well able to mimic its Lagrangian counterpart.

In this paper, we describe the construction of an efficient probabilistic parameterization that could be used in a coarse-resolution numerical model in which the variation of moisture is not properly resolved. An Eulerian model using a coarse-grained field on a grid cannot properly resolve regions of saturation—in which condensation occurs—that are smaller than the grid boxes. Thus, in the absence of a parameterization scheme, either the grid box must become saturated or condensation will be underestimated. On the other hand, in a stochastic Lagrangian model of moisture transport, trajectories of parcels tagged with humidity variables are tracked, and small-scale moisture variability can be retained; however, explicitly implementing such a scheme in a global model would be computationally prohibitive. One way to introduce subgrid-scale saturation into an Eulerian model is to assume the humidity within a grid box has a probability distribution. To close the problem, this distribution is conventionally determined by relating the required subgrid-scale properties of the flow to the grid-scale properties using a turbulence closure. Here, instead, we determine an assumed probability distribution by using the statistical moments from a stochastic Lagrangian version of the system. The stochastic system is governed by a Fokker–Planck equation, and we use that, rather than explicitly following the moisture parcels, to determine the parameters of the assumed distribution. We are thus able to parameterize subgrid-scale condensation in an Eulerian model in a computationally efficient and theoretically well-founded way. In two idealized advection–condensation problems, we show that a coarse Eulerian model with the subgrid parameterization is well able to mimic its Lagrangian counterpart.

Atmospheric water vapour is an essential ingredient of weather and climate. The key features of its distribution can be represented by kinematic models which treat it as a passive scalar advected by a prescribed flow and reacting through condensation. Condensation acts as a sink that maintains specific humidity below a prescribed, space-dependent saturation value. To investigate how the interplay between large-scale advection, small-scale turbulence and condensation controls moisture distribution, we develop simple kinematic models which combine a single circulating flow with a Brownian-motion representation of turbulence. We first study the drying mechanism of a water-vapour anomaly released inside a vortex at an initial time. Next, we consider a cellular flow with a moisture source at a boundary. The statistically steady state attained shows features reminiscent of the Hadley cell such as boundary layers, a region of intense precipitation and a relative humidity minimum. Explicit results provide a detailed characterization of these features in the limit of strong flow.

We show that the nature of the scaling behavior can be revealed by studying the conditional structure functions evaluated at given values of the locally averaged thermal dissipation rate. These conditional structure functions have power-law dependence on the value of the locally averaged thermal dissipation rate, and such dependence for the Bolgiano-Obukhov scaling is different from the other scaling behaviors. Our analysis of experimental measurements verifies the power-law dependence and reveals the Bolgiano-Obukhov scaling behavior at the center of the bottom plate of the convection cell.

We show that some statistical properties of forced two-dimensional turbulence have an important sensitivity to the form of large-scale dissipation, which is required to damp the inverse cascade. We consider three models of large-scale dissipation: linear “Ekman” drag, nonlinear quadratic drag, and scale-selective hypo-drag that damps only low-wavenumber modes. In all cases, the statistically steady vorticity field is dominated by almost axisymmetric vortices, and the probability density function of vorticity is non-Gaussian. However, in the case of linear and quadratic drag, we find that the velocity statistics is close to Gaussian, with non-negligible contribution coming from the background turbulent flow. On the other hand, with hypo-drag, the probability density function of velocity is non-Gaussian and is predominantly determined by the properties of the vortices. With hypo-drag, the relative positions of the vortices and the exponential distribution of the vortex extremum are important factors responsible for the non-Gaussian velocity statistics.