Eulerian and Lagrangian means in rotating, magnetohydrodynamic flows II. Braginsky’s nearly axisymmetric dynamo

The Hybrid Euler–Lagrange (HEL) approach has been usefully applied to weakly dissipative systems characterised by waves riding on mean flow. Soward (Phil. Trans. R. Soc. Lond. A 1972, 272, 431) showed how the HEL-formulation could elucidate remarkable features of the nearly axisymmetric large magnetic Reynolds number dynamo of Braginsky (JETP 1964, 47, 1084). Since Braginsky’s treatment of the nearly axisymmetric dynamo relies on azimuthal averages, those can only be taken when the azimuth is a coordinate direction. In that respect, the unified derivation and presentation of the HEL-equations governing rotating magnetohydrodynamic convective flows, as later reviewed and extended by Roberts and Soward (Geophys. Astrophys. Fluid Dyn. 2006, 100, 457), suffer the shortcoming that it was developed relative to rectangular Cartesian coordinates. Here we undertake those modifications needed to transform the rectangular Cartesian coordinate formulation into cylindrical polar coordinates. Being a Lagrangian description, application of the HEL-method means that the variables used, dependent on coordinates x, do not describe conditions at the position P: x but on conditions elsewhere at some displaced position PL: xL(x, t) = x + ξ (x, t), generally dependent on time t. To address this issue Soward and Roberts (J. Fluid Mech. 2010, 661, 45) invoked an idea pioneered by Moffatt (J. Fluid Mech. 1986, 166, 359), whereby the point x is dragged to xL(x, t) by a “fictitious steady flow” η(x, t) in a unit of “fictitious time”. This is the “Lie dragging” technique of general tensor calculus, which we apply here to the HEL-equations governing Braginsky’s nearly axisymmetric dynamo. We consider the “effective-variables” introduced by Braginsky, appropriate for small displacement ξ , and show that η, rather than ξ , is their natural expansion variable. As well as revisiting Braginsky’s kinematic dynamo, we reassess the hydromagnetic extensions of Tough and Roberts (Phys. Earth Planet. Inter. 1968, 1, 288)

Frequency staircases in narrow-gap spherical Couette flow

This is the author accepted manuscript. The final version is available from Taylor & Francis via the DOI in this record.Recent studies of plane parallel flows have emphasised the importance of finite-amplitude self-sustaining processes for the existence of alternative non-trivial solutions. The idea behind these mechanisms is that the motion is composed of distinct structures that interact to self-sustain. These solutions are not unique and their totality form a skeleton about which the actual realised motion is attracted. Related features can be found in spherical Couette flow between two rotating spheres in the limit of narrow-gap width. At lowest order the onset of instability is manifested by Taylor vortices localised in the vicinity of the equator. By approximating the spheres by their tangent cylinders at the equator, a critical Taylor number based on the ensuing cylindrical Couette flow problem would appear to provide a lowest order approximation to the true critical Taylor number. At next order, the latitudinal modulation of their amplitude a satisfies the complex Ginzburg-Landau equation (CGLe) ∂a/∂t=(λ+ix)a+∂2a/∂x2−|a|2a, ∂a/∂t=(λ+ix)a+∂2a/∂x2-|a|2a, where −x-x is latitude scaled on the modulation length scale, t is time and λλ is proportional to the excess Taylor number. The amplitude a governed by our CGLe is linearly stable for all λλ but possesses non-decaying nonlinear solutions at finite λλ, directly analogous to plane Couette flow. Furthermore, whereas the important balance ∂a/∂t=ixa∂a/∂t=ixa suggests that the Taylor vortices ought to propagate as waves towards the equator with frequency proportional to latitude, the realised solutions are found to exist as pulses, each locked to a discrete frequency, of spatially modulated Taylor vortices. Collectively they form a pulse train. Thus the expected continuous spatial variation of the frequency is broken into steps (forming a staircase) on which motion is dominated by the local pulse. A wealth of solutions of our CGLe have been found and some may be stable. Nevertheless, when higher-order terms are reinstated, solutions are modulated on a yet longer length scale and must evolve. So, whereas there is an underlying pulse structure in the small but finite gap limit, motion is likely to be always weakly chaotic. Our CGLe and its solution provides a paradigm for many geophysical and astrophysical flows capturing in minimalistic form interaction of phase mixing ixaixa, diffusion ∂2a/∂x2∂2a/∂x2 and nonlinearity |a|2a|a|2a.This paper was inspired by AMS’ attendance (19–21 March 2014) of the LMS Society Conference “Nonlinear stability theory: from weakly nonlinear theory to the verge of turbulence” to celebrate the 85th birthday of Professor J.T. Stuart. AMS gained further perspectives from attending (24–27 March 2014) the KITP program “Wave-flow interaction in geophysics, climate, astrophysics, and plasmas” at UCSB, where this research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. We are grateful to the referees for their helpful comments

Non-local effects in the mean-field disc dynamo. II. Numerical and asymptotic solutions

The thin-disc global asymptotics are discussed for axisymmetric mean-field dynamos with vacuum boundary conditions allowing for non-local terms arising from a finite radial component of the mean magnetic field at the disc surface. This leads to an integro-differential operator in the equation for the radial distribution of the mean magnetic field strength, $Q(r)$ in the disc plane at a distance $r$ from its centre; an asymptotic form of its solution at large distances from the dynamo active region is obtained. Numerical solutions of the integro-differential equation confirm that the non-local effects act similarly to an enhanced magnetic diffusion. This leads to a wider radial distribution of the eigensolution and faster propagation of magnetic fronts, compared to solutions with the radial surface field neglected. Another result of non-local effects is a slowly decaying algebraic tail of the eigenfunctions outside the dynamo active region, $Q(r)\sim r^{-4}$, which is shown to persist in nonlinear solutions where $\alpha$-quenching is included. The non-local nature of the solutions can affect the radial profile of the regular magnetic field in spiral galaxies and accretion discs at large distances from the centre.Comment: Revised version, as accepted; Geophys. Astrophys. Fluid Dyna

On the magnetic fields generated by experimental dynamos

We review the results obtained by three successful fluid dynamo experiments and discuss what has been learnt from them about the effect of turbulence on the dynamo threshold and saturation. We then discuss several questions that are still open and propose experiments that could be performed to answer some of them.Comment: 40 pages, 13 figure

Effects of alirocumab on types of myocardial infarction: insights from the ODYSSEY OUTCOMES trial

Aims  The third Universal Definition of Myocardial Infarction (MI) Task Force classified MIs into five types: Type 1, spontaneous; Type 2, related to oxygen supply/demand imbalance; Type 3, fatal without ascertainment of cardiac biomarkers; Type 4, related to percutaneous coronary intervention; and Type 5, related to coronary artery bypass surgery. Low-density lipoprotein cholesterol (LDL-C) reduction with statins and proprotein convertase subtilisin–kexin Type 9 (PCSK9) inhibitors reduces risk of MI, but less is known about effects on types of MI. ODYSSEY OUTCOMES compared the PCSK9 inhibitor alirocumab with placebo in 18 924 patients with recent acute coronary syndrome (ACS) and elevated LDL-C (≥1.8 mmol/L) despite intensive statin therapy. In a pre-specified analysis, we assessed the effects of alirocumab on types of MI. Methods and results  Median follow-up was 2.8 years. Myocardial infarction types were prospectively adjudicated and classified. Of 1860 total MIs, 1223 (65.8%) were adjudicated as Type 1, 386 (20.8%) as Type 2, and 244 (13.1%) as Type 4. Few events were Type 3 (n = 2) or Type 5 (n = 5). Alirocumab reduced first MIs [hazard ratio (HR) 0.85, 95% confidence interval (CI) 0.77–0.95; P = 0.003], with reductions in both Type 1 (HR 0.87, 95% CI 0.77–0.99; P = 0.032) and Type 2 (0.77, 0.61–0.97; P = 0.025), but not Type 4 MI. Conclusion  After ACS, alirocumab added to intensive statin therapy favourably impacted on Type 1 and 2 MIs. The data indicate for the first time that a lipid-lowering therapy can attenuate the risk of Type 2 MI. Low-density lipoprotein cholesterol reduction below levels achievable with statins is an effective preventive strategy for both MI types.For complete list of authors see http://dx.doi.org/10.1093/eurheartj/ehz299</p

Effect of alirocumab on mortality after acute coronary syndromes. An analysis of the ODYSSEY OUTCOMES randomized clinical trial

Background: Previous trials of PCSK9 (proprotein convertase subtilisin-kexin type 9) inhibitors demonstrated reductions in major adverse cardiovascular events, but not death. We assessed the effects of alirocumab on death after index acute coronary syndrome. Methods: ODYSSEY OUTCOMES (Evaluation of Cardiovascular Outcomes After an Acute Coronary Syndrome During Treatment With Alirocumab) was a double-blind, randomized comparison of alirocumab or placebo in 18 924 patients who had an ACS 1 to 12 months previously and elevated atherogenic lipoproteins despite intensive statin therapy. Alirocumab dose was blindly titrated to target achieved low-density lipoprotein cholesterol (LDL-C) between 25 and 50 mg/dL. We examined the effects of treatment on all-cause death and its components, cardiovascular and noncardiovascular death, with log-rank testing. Joint semiparametric models tested associations between nonfatal cardiovascular events and cardiovascular or noncardiovascular death. Results: Median follow-up was 2.8 years. Death occurred in 334 (3.5%) and 392 (4.1%) patients, respectively, in the alirocumab and placebo groups (hazard ratio [HR], 0.85; 95% CI, 0.73 to 0.98; P=0.03, nominal P value). This resulted from nonsignificantly fewer cardiovascular (240 [2.5%] vs 271 [2.9%]; HR, 0.88; 95% CI, 0.74 to 1.05; P=0.15) and noncardiovascular (94 [1.0%] vs 121 [1.3%]; HR, 0.77; 95% CI, 0.59 to 1.01; P=0.06) deaths with alirocumab. In a prespecified analysis of 8242 patients eligible for ≥3 years follow-up, alirocumab reduced death (HR, 0.78; 95% CI, 0.65 to 0.94; P=0.01). Patients with nonfatal cardiovascular events were at increased risk for cardiovascular and noncardiovascular deaths (P<0.0001 for the associations). Alirocumab reduced total nonfatal cardiovascular events (P<0.001) and thereby may have attenuated the number of cardiovascular and noncardiovascular deaths. A post hoc analysis found that, compared to patients with lower LDL-C, patients with baseline LDL-C ≥100 mg/dL (2.59 mmol/L) had a greater absolute risk of death and a larger mortality benefit from alirocumab (HR, 0.71; 95% CI, 0.56 to 0.90; Pinteraction=0.007). In the alirocumab group, all-cause death declined wit h achieved LDL-C at 4 months of treatment, to a level of approximately 30 mg/dL (adjusted P=0.017 for linear trend). Conclusions: Alirocumab added to intensive statin therapy has the potential to reduce death after acute coronary syndrome, particularly if treatment is maintained for ≥3 years, if baseline LDL-C is ≥100 mg/dL, or if achieved LDL-C is low. Clinical Trial Registration: URL: https://www.clinicaltrials.gov. Unique identifier: NCT01663402