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About TogetHER
TogetHER's 2019 Year in Review
The Global Burden of Cervical Cancer
Cervical Cancer: the Solutions
The TogetHER Interviews
Cervical Cancer Briefs for Global Fund programs
Cervical Cancer Grants Program
Our Work on Cervical Cancer
TogetHER's Partners
The TogetHER Interview: Nikolaj Gilbert, PATH President and CEO
The impact of COVID-19 on global cervical cancer prevention efforts has been wide-ranging, with unclear long-term ramifications. Our new TogetHER Interview series provides an opportunity for key leaders in global cervical cancer control to discuss current challenges and provide perspective on where the field needs to go from here.
This installment of the TogetHER Interview series features PATH President & CEO Nikolaj Gilbert.
Header photo: PATH
How have you and your organization been impacted by and adapted to COVID-19?
PATH has not been immune to the challenges that COVID-19 has dealt all of us. As a public health organization, we know what it means to respond to an epidemic, but we've never had to do it at this scale. We have adjusted our priorities to aid the global effort, and we have shifted how we do our work. For most of us that means working remotely while caring for children and other family members. We have experienced loss of life and are facing an uncertain future. It has not been easy.
And yet despite these challenges we remain committed to our mission. I have seen incredible resilience from our global team and some really important work accomplished. Our team in Vietnam helped set up digital tracking systems for the local response. Our team in Senegal has successfully pivoted tools and systems used for Ebola and malaria to aid with COVID. And across the board we've been thinking about how we can make changes that support the COVID response that also make health systems stronger long-term — while ensuring decades of progress fighting other diseases is not undone. I am so proud to be part of PATH and to work alongside these amazing people.
What concerns you most about women's health in low-resource countries during the pandemic? What is your organization doing about it?
The devastation of an infectious disease outbreak cannot be defined by its morbidity and mortality alone. Dwindling resources, decreases in access to health care services, and economic instability create just as much disruption in the health of communities — and exacerbate existing inequities.
This is true for women's health, not just in some places but everywhere. Limited access to primary health care, including sexual and reproductive health services and maternity care, limited contraception supply, and increased risk of violence are just a few of the ways the pandemic is making it harder for women to be healthy.
But there are things we are doing right now to alleviate some of those challenges. As governments respond to the COVID-19 pandemic, PATH is engaging with ministries of health to ensure that sexual and reproductive health and family planning are prioritized as essential health services. Additionally, we are tracking disruptions in family planning service delivery to inform our programming and response. We're also re-thinking the way health care is delivered and relying on remote consultations and at-home testing, treatment, and self-injected contraception. I hope we find solutions that don't just work now, but make access easier in the future too.
"As governments respond to the COVID-19 pandemic, PATH is engaging with ministries of health to ensure that sexual and reproductive health and family planning are prioritized as essential health services."
The Health Emergency Operations Center in Senegal, which PATH helped the government establish after the 2014 Ebola outbreak. Photo: PATH
How do you see cervical cancer prevention – the HPV vaccine and screen-and-treat – coming back after the pandemic? What can we do now to prepare for the resumption of services?
Cervical cancer is a preventable disease, but only if women are vaccinated against HPV, routinely screened, and treated when necessary. This standard of practice isn't established everywhere. Some countries have taken steps to ensure these services are available to all women, but those foundational measures are at risk.
For instance, PATH has partnered with local governments to create HPV vaccination programs in schools. This is the most effective way to reach adolescent girls, who should receive the vaccine between the ages of ages 9-14. The pandemic has led to massive school closures globally, and a marked decline (and in some cases halt) in vaccination. This disruption will likely have repercussions long term.
Screening programs have also been curtailed. Many of PATH's collaborators have re-allocated their screening personnel to the response effort, and staff have been unable to work because of lockdown. Cancer screening has been effectively put on hold.
We must resume services as soon as possible to avoid a lag in diagnosis and treatment. This is true for cervical cancer, but also other essential health services. Testing and safeguards from COVID-19 infection, measures to avoid overcrowding, and virtual consultations paired with self-collected sampling for testing are part of the solution.
"The pandemic has led to massive school closures globally, and a marked decline (and in some cases halt) in vaccination. This disruption will likely have repercussions long term."
Veronica Nanyombi (left), and Sarah Kemirembe (right), students at Nakasongola Primary School in Uganda, received the Human papillomavirus (HPV) vaccine through PATH's HPV vaccine project. Photo: PATH/Will Boase
What is something you have learned during Covid-19 that makes you hopeful?
I joined PATH only a few weeks before the first cases of COVID-19 appeared. I came from UNOPS, and previously from the pharmaceutical and finance industries, so I was well prepared, but new to the global public health sector. It has been quite an onboarding experience!
I think what has been most meaningful to me is the tenacity of this sector. Even when things are as hard as they have ever been, we are saying, "We must do better." We're reckoning with the history of colonialism and racism in our sector. We're becoming more efficient. We are taking steps to hold ourselves accountable and set the bar higher. I believe we will get past this and we will have used this as an opportunity to make changes that were long overdue. The people in PATH, the people responding to COVID, the people continuing to provide the essential services everyone needs to stay healthy. They give me hope, and I'm grateful.
Learn more about PATH's work on HPV vaccines and cervical cancer.
Interested in reading more TogetHER Interviews with leaders in the global cervical cancer response? Click here for the full list.
Nikolaj Gilbert is president and chief executive officer of PATH and managing director of PATH's Swiss subsidiary, Foundation for Appropriate Technology in Health. He brings to his roles more than 20 years of international experience as a leader, strategist, and director of complex partnerships. With a proven track record in both the private and public sectors, his expertise spans health, peacebuilding, humanitarian aid, and development solutions.
Before joining PATH in 2020, Mr. Gilbert served as director of partnerships for the United Nations Office of Project Services. In that position, he cultivated a deep understanding of the development ecosystem and of health inequity's underlying causes—from those rooted in global systems and markets, to those at local levels.
Based on his development sector and health industry expertise, Mr. Gilbert has an intimate knowledge of health agendas and priorities across donor and partner countries. He has demonstrated his ability to drive change at scale by building new programs and diversified funding models.
Previously, Mr. Gilbert served as a director for Novo Nordisk, where he developed expertise across corporate strategy, public-private partnership, business development, global marketing, and finance. He began his career at Accenture, a global management consulting firm, and has also worked with management research at INSEAD, a business school.
Mr. Gilbert is a Danish citizen and holds advanced degrees in business administration from Copenhagen Business School in Denmark and Ivey Business School in Canada.
Cervical Cancer Overview
TogetHER is fiscally sponsored by Panorama Global, a 501(c)(3), non-profit organization.


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The labelling perspective associated with Berger and Luckmann focuses on the processes by which some behaviours and types of people become marked out for social disapproval – targeted by the wider society as different and requiring some form of social response. Its virtue is that it challenges conventional assumptions that social problems exist 'out there' as obvious and commonly understood facts. Berger and Luckmann's perspective stresses the importance of language in shaping how we define, understand and respond to social problems by drawing attention to the role of labelling. However, the perspective is less helpful in dealing with the question of why some conditions become identified as problems and why some sorts of social constructions are in widespread use. One of the ways in which the labelling perspective has been developed is through linking social construction to issues of social interests, power and ideologies.
In this view, societies are characterised by patterns of inequality between social groups, for instance between different classes, between men and women, or between different ethnic groups. Groups in dominant positions in society will try to use sets of ideas that legitimate existing arrangements – and their positions within them. For example, groups or classes who control a large share of a society's wealth are likely to spread ideas about the necessity or even desirability of these patterns of economic inequality. They may refer to the fact that wealth reflects social virtue ('breeding') or accomplishment (the reward for 'risk-taking'). They may argue that inequality is necessary to encourage everyone to strive to be successful. They may argue that wealth is a responsibility held in trust for future generations. Such sets of ideas that legitimate advantageous positions tend to be called ideologies. Similar ideologies have been used to justify the inequalities between men and women (men are stronger, better at thinking abstractly; women are better at caring or need to conserve their resources for the demands of childbirth, and so on). As both these examples suggest, one common feature of such ideologies is that they claim that existing social arrangements are legitimate because they are 'natural' (and therefore both desirable and unavoidable). Each structural pattern of inequality is likely to create interests (in preserving or changing the existing pattern) and these interests are likely to provide the foundation for ideologies which seek to legitimate or challenge these inequalities.
This is the most simple view of ideology: as a set of ideas which attempt to legitimate (or challenge) social position and inequality. Even in this most simple form, however, we can see ways in which it can link to and develop the labelling approach to social construction.
Let us return to the issue of poverty. How might poverty be defined in ways which would reinforce or challenge the position of the powerful?
We suggest that it might be in the interests of those with wealth to deny that poverty is a social problem (or even that it exists). If poverty can be explained as being the result of 'sad, bad or mad' people – those who are workshy, feckless or incompetent – then poverty is nothing to do with the social arrangements that make some people wealthy. Alternatively, poverty can be explained as a necessary, if unfortunate, by-product of a system that creates wealth. We might even go so far as to say that the idea of poverty is itself tolerable within such an ideology because it separates out the 'poor' as a social problem, rather than revealing the underlying structure of inequality. In distinguishing between the normal and the deviant, the wider inequalities and differences of social interest are concealed. On the other hand, if poverty were solely blamed either on the behaviour of the rich or on the operation of an unfair or unjust economic system, it might underpin a political ideology which aimed to challenge the powerful in society.
Ideologies are also likely to be involved in defining who and what are social problems. Most obviously, those who challenge existing social arrangements are likely to be defined as troublemakers, agitators or rabble-rousers who aim to cause unnecessary social disruption and disaffection. Alternatively, such challenges may be dismissed as Utopian, unworldly or even unnatural because they claim that other ways of arranging the social world are possible. Beyond this, however, there are ways in which specific social conditions or patterns of behaviour might be identified as social problems because they run counter to the interests of dominant groups. For example, some social scientists have suggested that the interests of a capitalist class mean that it needs a healthy and reliable workforce. In the period of industrialisation, therefore, indolence and insobriety came to be defined as social problems because they threatened to disrupt patterns of work and the norm of a sober and industrious worker (see, for example, Clarke and Critcher, 1985). Equally, it has been argued that the industrialists of Britain came to take a growing interest in the 'health of the nation' at the end of the nineteenth century, when they became concerned that ill-health and malnutrition made the British working class less fit and less competitive by comparison with other industrialising nations. Social problems here are the result of interests being threatened, and the definitions of those problems stem from the power of dominant groups to determine what counts as a problem.
In this view ideologies are functional. That is, they are geared to legitimating social positions. In principle, one can look behind ideologies to see what interests they protect and serve. They are also intended to mystify or conceal what society is really like by explaining away inequalities. So, for example, these legitimating ideologies may celebrate forms of equality (equality before the law, for instance) which distract attention from other areas of social life where inequality is rife. Anatole France referred ironically to 'the majestic equality of the French law, which forbids both rich and poor from sleeping under the bridges of the Seine' (quoted in CSE/NDC, 1977, pp. 24–5).
Previous 5.3 Summary
Next 6.2 Contesting ideologies

redpajama_set_name: RedPajamaCommonCrawl


18-39years (for Oc and General Candidates).
All the Applying candidates or applicants are going to pay the fee as Two Hundred Fifty Rupees. All the applicants have to pay the fee for every post go through the AP-ONLINE or e-seva centers between the dates on 16-February-2012 and 16-March-2012.
All the Eligible, Qualified and Interested Candidates are going to apply online at ap dsc official website as wwwwwW..apdsc.cgg.gov.in.
i have been trying for online application form for registering but could not open why please make web page available for us to fill up.


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**Nova is a library for running NodeJS code in the browser. Nova is written in coffeescript, deal with it.**

## Usage

Using nova is as simple as including nova.js on your web page. Please note that nova is a newborn baby and only supports certain NodeJS functionality. Obviously we can't run an HTTP server from IE6 and we can't initiate TCP connections from your Netscape. (or can we?)

## Support

nova currently comes with the following NodeJS modules:

```
events - 100%
util - 100%
assert - 100%
sys - 100%
readline - 85% (All console management functionality does nothing though)
fs - 50% (LocalStorage filesystem models need refinement)
http - 25% (Based on superagent AJAX which needs to go)
```

## Size

When developing libraries for the browser, size matters. nova is currently 4.3k minified and 1.1k gzipped. NodeJS functionality in under 10k? That's cool. For comparison, jQuery is currently 91.6k. You can always pick and choose which modules are included in nova so your size may be even smaller.

## Plugins

Not only does nova plan on mapping most NodeJS functionality to the browser, we also plan on extending it! This means modules for popular DOM manipulation libraries and other browser-only goodies. You can also easily write your own modules for nova (check the example folder) that extend or bring new functionality to it entirely.

## Examples

You can view further examples in the [example folder.](https://github.com/wearefractal/nova/tree/master/examples)

## Dependencies

As of now, nova only aims to be compliant with the latest browsers. A separate build will be available that is jammed packed with polyfills and shims at some point in the future. If you want nova to work with older browsers THIS MOMENT you can check out [this page by Modernizr](https://github.com/Modernizr/Modernizr/wiki/HTML5-Cross-browser-Polyfills) that has a massive list of shims/polyfills.
Developers will need coffee-script and uglify-js installed globally via npm to build nova and fusker to run the test server.

## Contributors

- [Contra](https://github.com/Contra)

## LICENSE

(MIT License)

Copyright (c) 2011 Fractal <contact@wearefractal.com>

Permission is hereby granted, free of charge, to any person obtaining
a copy of this software and associated documentation files (the
"Software"), to deal in the Software without restriction, including
without limitation the rights to use, copy, modify, merge, publish,
distribute, sublicense, and/or sell copies of the Software, and to
permit persons to whom the Software is furnished to do so, subject to
the following conditions:

The above copyright notice and this permission notice shall be
included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

redpajama_set_name: RedPajamaGithub


Vase with profile of a ladies head on each side. Britannia - Metal (Pewter and antimony alloy) Glass - crystal glass with flower cameo decor. Inspired with design WMF.
Decorative fruit stand in the form lady with art nouveau ornaments. Britannia - Metal (Pewter and antimony alloy). Plate - cut crystal glass with flower cameo decor. Inspired with design WMF.
Jardiniere with a profile of a ladies and art nouveau ornaments on each side. Britannia - Metal (Pewter and antimony alloy) Glass - cut crystal glass with optical decor. Inspired with design WMF.
Vase with a figure of lady and art nouveau ornaments on each side. Britannia - Metal (Pewter and antimony alloy) Glass - cut crystal glass. Inspired with design WMF.


Are Leases Helping or Hurting Deer Hunting?
This past deer season I harvested my largest buck to date – maybe the largest of my life – on a new property I leased in Kansas. The old warrior of a buck scored just over 175 inches and was a dream-come-true for a guy who cut his teeth hunting the heavily-pressured state lands of Pennsylvania. That said, I can't help but think about the last four years.
Why those four in particular? I had finally escaped my seven-year, two-degree tenure as a college student and was finally making some "real money." Like most recent grads, a fulltime salary sounds like a lot while you're attending or just coming out of school. But once you're married and dive off into a mortgage, car payments, student loans and then decide to have children, it seems to never be enough (usually isn't). Living responsibilities made additional funds for a hunting lease few and far between.
When I first graduated, my wife and I landed on a decent lease in central Kentucky, about 45 minutes from our house. She killed a great buck that year; we gained a lot of memories with that lease. Soon after, we moved to the big-buck producing state of Missouri for work related changes.
Probably before the boxes were unpacked, I was looking for a hunting lease. To my surprise I couldn't find a lease for less than $20 per acre. Insane! Being new to the area, we didn't know anyone that could give us a better lead than Craigslist or Google, so we hunted public land for the first year. But that soon turned into a second year, and then a third. Even this year, I only hunted public land when in Missouri.
Now don't get me wrong, I love the challenge. But I have to think that these high prices may be what are driving more and more hunters out of the sport. I mean hunter access is one of the top issues facing future generations, right?
On the other hand, the people engaging in these high-priced leases are often spending more money on things like food plots, treestands or blinds, and even equipment like tractors. These funds drive Pittman-Robertson which benefit not just game, but all wildlife and habitat conservation. Many are doing more on the lease to promote better and healthier native habitat and wildlife populations.
With much of the U.S. privately owned, it is critical for these lands to be managed appropriately by those accessing them. I know I put some extra cash into the Kansas lease and already have plans to expand the operation this year.
I guess it's a toss-up. For some, the lease prices will be the demise of their hunting career. For others, it will drive them to hunt public land, which can be very productive with the right homework. Lastly, it will likely lead to more money put towards products that will benefit the Pittman-Robertson funding and fuel wildlife conservation for future generations.
I'll end with this: Regardless of where you hunt or how you hunt, the reality is you do it because you love it. That should never fade. Sure, I'm still on "cloud 9" after shooting my buck this fall on the Kansas lease. But I can honestly say my best memories are those I made, and continue to make, beating the public ground because I know the amount of effort and skill it took to succeed.


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\section{Introduction}

Liquid metal turbulence is encountered in various situations, ranging from metallurgy \citep{Davidson} to the flow in the Earth's outer core \citep{Moffatt}, including laboratory experiments on magnetohydrodynamic (MHD) turbulence \citep{Alemany,Gallet}. Because their kinematic viscosity is much lower than their magnetic diffusivity, with magnetic Prandtl numbers $Pm$ typically in the range of $10^{-6}$ to $10^{-5}$, flows of liquid metals can be turbulent and still be characterized by a small magnetic Reynolds number: in most laboratory experiments on MHD turbulence, the kinetic Reynolds number is in the range $10^4-10^6$, with a magnetic Reynolds number rarely exceeding unity. At small enough scales, turbulence in Earth's outer core may also be considered as a low-$Rm$ flow.

Turbulence subject to a strong external magnetic field tends to become two-dimensional, the velocity field becoming weakly dependent on the coordinate along the applied magnetic field (hereafter denoted as the vertical axis by convention). Indeed, any flow that depends on the vertical coordinate shears the applied field and induces electrical currents. For low magnetic Reynolds number, such currents are rapidly damped through Ohmic dissipation and the flow structures that depend on the vertical are more damped than vertically-invariant eddies. This intuitive argument can be made more precise in the asymptotic limit of vanishing magnetic Reynolds number $Rm$, the so-called quasi-static approximation. In this limit the applied field simply appears as an additional Ohmic damping term in the  Navier-Stokes equation \citep{Sommeria, Knaepen}. This term is dominant for large $B_0$, and it depends on the second vertical derivative of the velocity: for infinite $B_0$ the bulk velocity field must be independent of the vertical.

For finite but strong external magnetic field $B_0$, one may ask the following questions:  does the bulk flow remain three-dimensional, with weak three-dimensional velocity fluctuations superposed on a strong two-dimensional flow, or does it become purely two-dimensional? How do the Ohmic and viscous dissipation rates compare for large $B_0$? Is there a scaling law relating the vertical and horizontal gradients of the velocity field? Or is there a critical value of $B_0$ above which the flow becomes completely independent of the vertical?

These central questions are strongly related to the scaling of the energy dissipation in MHD turbulence. Indeed, three dimensional hydrodynamic turbulence displays a dissipation anomaly: the 3D flow of a fluid with kinematic viscosity $\nu$ forced on a scale $\ell$ with typical velocity $U$ dissipates a finite power per unit mass $\epsilon \sim U^3/\ell$ as the Reynolds number $U \ell / \nu$ goes to infinity. By contrast, body-forced two-dimensional flows do not present such a singular zero-viscosity limit: in 2D the dissipated power vanishes for vanishing viscosity, $\ell$ and $U$ being kept constant. Because the MHD situation with strong external $B_0$ is an intermediate situation between 2D and 3D turbulence, an open question is whether it displays a dissipation anomaly.

In the laboratory, the influence of an external magnetic field on the bulk turbulence of a liquid metal was studied in various geometries, ranging from channel flows to confined turbulence \citep{Alemany, Gallet, Klein, Potherat2014}. These studies indicated a clear trend toward two-dimensionalization as the external field increases. Experiments  were also performed in shallow layers of liquid metal subject a strong uniform magnetic field to maximize the two-dimensionalization \citep{Sommeria86, GalletGAFD}. The corresponding flows displayed good agreement with the phenomenology of 2D turbulence.

Numerical simulations at unit magnetic Prandtl number have also displayed partial two-dimensionalization, with the formation of large-scale two-dimensional structures coexisting with small-scale three-dimensional fluctuations \citep{Alexakis2D,Bigot}. \citet{Zikanov} focused their numerical study on MHD turbulence in the quasi-static limit $Rm \to 0$, and they observed three regimes: for weak applied field, the flow is three-dimensional. For intermediate $B_0$, its behavior is intermittent, with long phases of quasi-2D motion interrupted by 3D bursts. For large $B_0$, the system settles into a purely 2D state, the amplitude of the $z$-dependent part of the velocity field decreasing exponentially in time. Although the forcing procedure is somewhat unrealistic (they rescale the total energy of the large-scale Fourier modes to a constant value at each time step), their simulations provide a clean example of MHD turbulence reaching a purely 2D stationary state provided the external field is strong enough. On the theoretical side, they consider a given two-dimensional large-scale cellular flow, and they show that it is absolutely stable with respect to vertically-dependent perturbations for large enough external field. In a subsequent paper \citep{Thess}, they study the linear stability of several other steady two-dimensional flows as a function of the strength of the external field. For finite Reynolds number, both the elliptical vortex and the vortex sheet are stable provided the applied field is strong enough. Although enlightening, these examples do not give an unambiguous answer to the central question asked above, because they do not consider the general setup of body-forced turbulence.

Insightful analytical results on the behavior of such body-forced MHD flows have been obtained by \citet{Potherat2003} in the vanishing-$Rm$ approximation: using rigorous methods from mathematical analysis, these authors provide upper bounds on the dimension of the attractor of MHD turbulent flows. For strong external magnetic field, the bound follows the same scaling as the one obtained (with the same method) for purely 2D flows. This indicates that the behavior of such low-$Rm$ MHD flows somewhat ``resembles" that of 2D flows, but it does not prove that the flow really is 2D: for instance, the bound holds even when the forcing is three-dimensional, a situation for which the flow cannot be purely two-dimensional.

In this paper we use rigorous analysis and estimates to answer the central question of exact two-dimensionalization. We consider an MHD flow driven by a horizontal body force and subject to a vertical magnetic field, in a domain that is periodic in the horizontal and bounded vertically by stress-free surfaces. Such boundary conditions are chosen to remove the boundary layers inherent to experimental realizations, and to allow for a purely two-dimensional velocity field. We stress the fact that the results presented in this paper carry over to the 3D periodic cube, as discussed in the final section. In the quasi-static ($Rm\to 0$) approximation, we prove that the flow becomes exactly 2D asymptotically in time, regardless of the initial condition, provided the interaction parameter $N$ is larger than a critical value $N_c(Re)$. We call this property {\it absolute two-dimensionalization}. We then consider the full-MHD system and study the evolution of vertically-dependent perturbations of infinitesimal amplitude. We show that such infinitesimal perturbations decrease provided the interaction parameter is larger than some $Re$-dependent threshold value, and the magnetic Reynolds number is lower some other $Re$-dependent threshold. We thus extend the domain in which we prove that the two-dimensional flow is an attractor, but at the price of considering infinitesimal 3D perturbations only. We therefore call this property {\it linear two-dimensionalization}.

In section \ref{section2} we describe the MHD system and recall some bounds on the viscous energy dissipation in hydrodynamic turbulence. Section \ref{section3} contains the proof of absolute two-dimensionalization for quasi-static MHD ($Rm \to 0$). Sections \ref{section4} and \ref{section5} deal with linear two-dimensionalization for the full MHD equations. In section \ref{section6} we summarize the different results obtained in this paper and discuss their implications for the energy dissipation rate of low-$Rm$ MHD turbulence. Some  technical aspects of the proofs are presented in three appendices.

The rigorous estimates of this study are based on Young's, H\"older's, Poincar\'e's, the Cauchy-Schwarz, and Ladyzhenskaya's inequalities. Young's inequality is simply the fact that $|ab|\leq \frac{1}{2c}a^2+\frac{c}{2}b^2$ for any $c>0$, where $c$ can be optimized to get the best possible upper bound. H\"older's, Poincar\'e's and the Cauchy-Schwarz inequalities are presented in a fluid dynamics context in \citet{Doering}, while Ladyzhenskaya's inequality is proven in \citet{Ladyzhenskaya} (Lemma 1, page 8).

\section{Body-forced magnetohydrodynamics and hydrodynamics\label{section2}}

\subsection{Magnetohydrodynamics in a cubic domain}

\begin{figure}
\begin{center}
\includegraphics[width=120 mm]{fig1.eps}
\end{center}
\caption{Flow of an electrically-conducting newtonian fluid subject to a uniform vertical magnetic field $B_0$. It is driven by a horizontal body-force $\textbf{f}$ that is independent of the vertical. We assume periodic boundary-conditions in the horizontal, and stress-free boundaries with large magnetic-permeability  at $z=0$ and $z=H$.}
\label{3Dbox}
\end{figure}

We consider the setup sketched in figure \ref{3Dbox}: an incompressible fluid of kinematic viscosity $\nu$, density $\rho$ and electrical conductivity $\sigma$ flows inside a domain $(x,y,z) \in \mathcal{D}=[0, L] \times [0,L] \times [0,H]$ with a Cartesian frame $(\textbf{e}_x,\textbf{e}_y,\textbf{e}_z)$. It is stirred by a time-independent divergence-free two-dimensional horizontal body-force $\textbf{f}(x,y)=(f_x,f_y,0)$ that is periodic on a scale $\ell$, an integer fraction of $L$. That is, ${\bf f}(x,y)=F {\boldsymbol \phi}(\frac{x}{\ell},\frac{y}{\ell})$, where ${\boldsymbol \phi}$  is periodic of period $1$ in each dimensionless variable and has an rms magnitude $1$. We refer to $F$ as the amplitude, and ${\boldsymbol \phi}$ as the shape of the force.

A homogeneous steady vertical magnetic field $B_0 \textbf{e}_z$ is applied to the system. We consider periodic boundary conditions in the horizontal directions ($x$ and $y$), and stress-free boundary conditions for the velocity field at the top and bottom boundaries. We further limit attention to the case that these two boundaries have very large magnetic permeability, so that the magnetic field is vertical at $z=0$ and $z=H$. Denoting $\textbf{u}(x,y,z,t)$ and $\textbf{B}(x,y,z,t)$ as the velocity and magnetic fields inside the fluid, the boundary conditions at the top and bottom boundaries of the domain are
\begin{equation}
\left.
\begin{array}{ll}
\partial_z u_x=0\,, \partial_z u_y=0\,, u_z=0 \\
B_x=0\,, B_y=0\,, \partial_z B_z=0
\end{array}
\right\} \mbox{at } z=0 \mbox{ and } z=H \, .
\label{BC}
\end{equation}

The magnetic field inside the fluid follows the induction equation, which using the decomposition $\textbf{B}(x,y,z,t)=B_0 \textbf{e}_z+ \textbf{b}(x,y,z,t)$ reads
\begin{equation}
\partial_t \textbf{b} + (\textbf{u} \cdot \boldsymbol{\nabla}) \textbf{b} - (\textbf{b} \cdot \boldsymbol{\nabla}) \textbf{u}= B_0 \partial_z \textbf{u} +\eta \Delta  \textbf{b}\, , \qquad \boldsymbol{\nabla}.\textbf{b} =0 \label{induction}
\end{equation}
where $\eta=1/(\mu_0 \sigma)$, with $\mu_0$ the magnetic permeability of vacuum. Without loss of generality, we assume $B_0>0$ in the following.

The velocity field follows the body-forced Navier-Stokes equation with the Lorentz force
\begin{equation}
\partial_t \textbf{u} +(\textbf{u} \cdot \boldsymbol{\nabla}) \textbf{u}=-\boldsymbol{\nabla}{p} + \nu \Delta \textbf{u} + \textbf{f}(x,y) + \frac{1}{\rho \mu_0} (\boldsymbol{\nabla} \times \textbf{b} )\times(B_0 \textbf{e}_z+ \textbf{b})\, , \qquad \boldsymbol{\nabla}.\textbf{u} =0 \, . \label{NS}
\end{equation}

From equations (\ref{induction}) and (\ref{NS}) we define the kinetic and magnetic Reynolds numbers using the root-mean-square velocity $U$, where the mean is performed over space and time:
\begin{equation}
Re=\frac{U \ell}{\nu}\, ,  \qquad Rm=\frac{U \ell}{\eta} \,.
\end{equation}
In the following, we focus on weak magnetic Reynolds numbers, $Rm \lower.7ex\hbox{$\;\stackrel{\textstyle<}{\sim}\;$} 1$, so that the dominant balance in (\ref{induction}) is
\begin{equation}
B_0 \frac{U}{\ell} \sim \frac{\eta b}{\ell^2}\, .
\end{equation}
One can define a third independent dimensionless quantity, the interaction parameter $N$, to evaluate the ratio between the Lorentz force and the inertial term in the Navier-Stokes equation:
\begin{equation}
N=\frac{\sigma B_0^2 \ell}{\rho U}=\frac{ B_0^2 \ell}{\rho \mu_0 \eta U} \, .
\end{equation}
$N$ is a dimensionless measure of the strength of the external field $B_0$.

This magnetohydrodynamic problem admits two-dimensional solutions of the form $\textbf{u}=\textbf{V}(x,y,t)$, $\textbf{b}=0$, where $\textbf{V}(x,y,t)$ satisfies a standard non-magnetic two-dimensional Navier-Stokes equation. Our goal is to prove that, for large enough interaction parameter, such motion is robust with respect to three-dimensional perturbations, i.e., the three-dimensional part of the velocity field decays in the long-time limit, and the system relaxes to a 2D flow.

\subsection{The quasi-static, or vanishing $Rm$ limit \label{QS}}

The quasi-static limit consists in going to the limit $Rm \to 0$ while the interaction parameter $N$ remains of order unity, before expanding the equations to lowest order in $Rm$. At this order, the induction equation (\ref{induction}) reduces to $\eta \Delta {\bf b} = - B_0 \partial_z {\bf u}$, i.e. the induced magnetic field follows the velocity field adiabatically. It is much weaker than the external field, with ${\bf b} \sim Rm B_0$. To lowest order in $Rm$, the Lorentz force is then $(\boldsymbol{\nabla} \times \textbf{b} )\times B_0 \textbf{e}_z / (\rho \mu_0)$, and the Navier-Stokes equation reads
\begin{equation}
\partial_t \textbf{u} +(\textbf{u} \cdot \boldsymbol{\nabla}) \textbf{u}=-\boldsymbol{\nabla}{p} + \nu \Delta \textbf{u} + \textbf{f}(x,y) - \beta \Delta^{-1}  \partial_{zz} {\bf u} \, , \qquad \boldsymbol{\nabla}.\textbf{u} =0 \, , \label{NSzeroRm}
\end{equation}
where $\Delta^{-1}$ is the inverse Laplacian operator, which can be thought of as a division by $k^2$ of each Fourier component, $k$ denoting the wavenumber. $\beta$ is the inverse Joule time scale corresponding to magnetic damping, $\beta=\sigma B_0^2/\rho =B_0^2/(\rho \mu_0 \eta)$. It is related to the interaction parameter through $N=\beta \ell / U$. The main convenience of the quasi-static equation (\ref{NSzeroRm}) is that it involves the velocity field only.

\subsection{The energy dissipation of 2D and 3D turbulence}

As mentioned in the introduction, the central challenge of the analysis is to establish the stability of some vertically-invariant flow ${\bf V}(x,y,t)$ with respect to vertically-dependent perturbations ${\bf v}(x,y,z,t)$, without knowing the details of the base flow ${\bf V}$: it is typically a turbulent solution of a two-dimensional body-forced Navier-Stokes system. Instead of a precise knowledge of ${\bf V}(x,y,t)$, for the proofs we make use of rigorous bounds on the time-averaged energy dissipation for solutions of the body-forced 2D and 3D Navier-Stokes equations. The corresponding bounds on the gradients of ${\bf V}$ are sufficient to establish its stability with respect to 3D perturbations. We now recall these bounds, committing the proofs to an appendix.

For hydrodynamic turbulence in a 3D periodic domain, \citet{Doering} derived a bound on the time-averaged viscous energy dissipation of any solution ${\bf u}$ to the body-forced Navier-Stokes equation, i.e., equation (\ref{NS}) without the Lorentz force. The method consists in bounding the amplitude of the body-force, and the power it inputs, in terms of the root-mean-square velocity $U$. We prove in appendix \ref{appbound} that the bound easily carries over to the present magnetohydrodynamic system in the quasi-static limit, i.e., when it is governed by (\ref{NSzeroRm}). In the following, the  constants with subscripts $(c_i)_{i\in\mathbb{N}}$ are dimensionless positive numbers that depend on the {\it shape} of the forcing function only, not on its amplitude or length scale $\ell$. By contrast, the undecorated $c$ denotes the generic positive $O(1)$ constant that does not depend on any parameter of the problem. With these notations, the bound reads

\begin{equation}
\left<  \| \boldsymbol{\nabla} \textbf{u}  \|_{2}^2 \right>   \leq \frac{U^3 L^2 H}{\ell \nu} \left( \frac{c_1}{Re} +c_2 \right) \, ,
\label{DF}
\end{equation}
where $\| \dots \|_2^2$ is the standard $L_2$ norm over $\mathcal{D}$, $\left< \dots \right>$ denotes time-average, and $c_1$ and $c_2$ depend only on the shape of the forcing function ${\bf f}$.

Bounds can also be obtained for solutions of the 2D Navier-Stokes equation, i.e., for vertically-invariant horizontal velocity fields satisfying equation (\ref{NS}). Denoting the vorticity as $\omega$, \citet{Alexakis} derived a bound on time-averaged dissipation rate of enstrophy $\| \omega \|_2^2$ in 2D. For a time-independent forcing function, and using our notations, their equation (19) translates into
\begin{eqnarray}
 \left< \| \boldsymbol{\nabla} \omega \|_2^2 \right> \leq \frac{H L^2 U^3}{\nu \ell^3} \left( c_3 + \frac{c_4}{Re} \right) \, , \label{2Dbound}\\
\end{eqnarray}
where $c_3$ and $c_4$ are again dimensionless constants that depend only on the shape of ${\bf f}$, and the prefactor $H$ arises from $\| \dots \|_2$ being the $L_2$ norm in 3D.

This bound for vertically-invariant flows can be further reduced if the forcing is such that $\boldsymbol{\nabla}^2 {\bf f}$ is proportional to ${\bf f}$. These harmonic forcings are sometimes called ``single-mode", or Kolmogorov forcings. For such forcings an improved bound on the enstrophy dissipation rate is
\begin{equation}
\left< \| \boldsymbol{\nabla} \omega \|_2^2 \right> \leq c \frac{U^2 L^2 H}{\ell^4} \, . \label{2DboundKolmogorov}
\end{equation}

\section{Absolute two-dimensionalization for vanishing magnetic Reynolds number\label{section3}}

\subsection{Evolution equation for the kinetic energy of the three-dimensional part of the velocity field\label{StratAbs}}

We consider the evolution of an initially 3D velocity field under equation (\ref{NSzeroRm}). Define the vertical average of a field ${\bf h}$ as
\begin{equation}
\bar{{\bf h}}(x,y,t) = \frac{1}{H} \int_{z=0}^{z=H}  {\bf h}(x,y,z,t) \, \mathrm{d}z \, ,
\end{equation}
and decompose the velocity field into
\begin{equation}
\textbf{u}=\textbf{V}(x,y,t)+\textbf{v}(x,y,z,t) \mbox{, with } \textbf{V}=\bar{\textbf{u}} \mbox{, and } \textbf{v}=\textbf{u}-\textbf{V} \, .
\label{decomp}
\end{equation}
In the following we call ${\bf V}$ (resp. ${\bf v}$) the two-dimensional (resp. three-dimensional) part of the velocity field. Our goal is to find a criterion that insures the decay of ${\bf v}$. One can obtain the evolution equation for the two-dimensional part of the velocity field by vertically-averaging the Navier-Stokes equation (\ref{NSzeroRm}),
\begin{equation}
\partial_t \textbf{V} +(\textbf{V} \cdot \boldsymbol{\nabla}) \textbf{V} +\overline{(\textbf{v} \cdot \boldsymbol{\nabla}) \textbf{v}}=-\boldsymbol{\nabla}{\bar{p}} + \nu \Delta \textbf{V} + \textbf{f}(x,y) \, .
\label{eqV}
\end{equation}
By subtracting this equation to the Navier-Stokes equation, we get the evolution equation for the three-dimensional part of the flow:
\begin{equation}
\partial_t \textbf{v} +(\textbf{V} \cdot \boldsymbol{\nabla}) \textbf{v} +(\textbf{v} \cdot \boldsymbol{\nabla}) \textbf{V} +(\textbf{v} \cdot \boldsymbol{\nabla}) \textbf{v} - \overline{(\textbf{v} \cdot \boldsymbol{\nabla}) \textbf{v}}=-\boldsymbol{\nabla}(p-\bar{p}) + \nu \Delta \textbf{v}  - \beta \Delta^{-1}  \partial_{zz} \textbf{v}  \, .
\label{eqv}
\end{equation}
The incompressibility constraint imposes $\boldsymbol{\nabla} \cdot \textbf{V} = 0$ and $\boldsymbol{\nabla} \cdot \textbf{v} = 0$. To get the evolution equation for the kinetic energy of the 3D part of the velocity field, one takes the dot product of (\ref{eqv}) with $\textbf{v}$ and integrates over the domain. Some integrations by parts using the incompressibility constraint lead to
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{1}{2} \|\textbf{v}\|_2^2 \right) = \mathcal{H}\{\textbf{v},t\}  ,
\label{eqv2}
\end{equation}
with $\mathcal{H}$ a time-dependent quadratic functional of the field $\textbf{v}$,
\begin{equation}
\mathcal{H}\{\textbf{v},t\} = - \int_\mathcal{D} \textbf{v} \cdot (\boldsymbol{\nabla} \textbf{V}) \cdot \textbf{v} \, \mathrm{d}^3 \textbf{x} - \nu \|\boldsymbol{\nabla} \textbf{v}\|_2^2 - \beta \|\boldsymbol{\nabla}^{-1}  \partial_z{\textbf{v}} \|_2^2\, ,
\label{defH}
\end{equation}
where we introduced the notation $\boldsymbol{\nabla}^{-1}=\boldsymbol{\nabla} \Delta^{-1}$

The strategy is the following: we seek a function $\lambda_1(t)$ such that
\begin{itemize}
\item For all divergence-free fields $\textbf{v}$ with zero vertical average ($\bar{\textbf{v}}=\textbf{0}$) satisfying the boundary conditions (\ref{BC}),
\begin{equation}
\mathcal{H}\{\textbf{v},t\} \leq \lambda_1(t) \| \textbf{v} \|_2^2 .
\label{ineqlambda}
\end{equation}
\item The long-time average of $\lambda_1$ is negative, i.e. $\left< \lambda_1 \right> <0$.
\end{itemize}

Then the kinetic energy in the three-dimensional part of the velocity field decays at least exponentially in time. The velocity field then becomes two-dimensional asymptotically in the long-time limit, regardless of the amplitude of the initial three-dimensional perturbation ${\bf v}$. In the following, we call this property of the system{ \it absolute two-dimensionalization}. Indeed, our approach for proving two-dimensionalization is very similar to proving absolute stability of some velocity field $\textbf{V}$ to perturbations $\textbf{v}$. However, it is conceptually distinct from a standard absolute stability analysis: in relation (\ref{defH}), $\textbf{V}(x,y,t)$ is a possibly turbulent velocity field that satisfies equation (\ref{eqV}), which includes an undetermined Reynolds stress term.

\subsection{A sufficient condition for two-dimensionalization}

To find a function $\lambda_1$ that satisfies (\ref{ineqlambda}), we need to bound the first term on the right-hand-side of (\ref{defH}). This task is performed in appendix \ref{appendix1}, where we use Ladyzhenskaya's inequality on the $L_4$ norm.  We obtain
\begin{eqnarray}
 \left| \int_\mathcal{D} \textbf{v} \cdot (\boldsymbol{\nabla} \textbf{V}) \cdot \textbf{v} \, \mathrm{d}^3 \textbf{x} \right| & \leq &  c \left( \frac{ \| \boldsymbol{\nabla} \textbf{V} \|_2}{L \sqrt{H}} + \frac{\| \boldsymbol{\nabla} \textbf{V} \|_2^2}{\nu H}  \right) \| {\bf v} \|_2^2 + \frac{\nu}{2}  \| \boldsymbol{\nabla} {\bf v} \|_2^2 \, .
\end{eqnarray}
We bound the last term in (\ref{defH}) using Poincar\'e's inequality in the $z$ direction only,
\begin{equation}
- \beta \|\boldsymbol{\nabla}^{-1}  \partial_z{\textbf{v}} \|_2^2 = - \beta \| \partial_z \boldsymbol{\nabla}^{-1}  {\textbf{v}} \|_2^2 \leq - \beta \frac{\pi^2}{H^2} \| \boldsymbol{\nabla}^{-1}  {\textbf{v}} \|_2^2 \, ,
\end{equation}
to get the following bound on the functional $\mathcal{H}$,
\begin{equation}
\mathcal{H}\{\textbf{v},t\} \leq c \left( \frac{ \| \boldsymbol{\nabla} \textbf{V} \|_2}{L \sqrt{H}} + \frac{\| \boldsymbol{\nabla} \textbf{V} \|_2^2}{\nu H}  \right) \| {\bf v} \|_2^2 - \beta \frac{\pi^2}{H^2} \| \boldsymbol{\nabla}^{-1}  {\textbf{v}} \|_2^2 - \frac{\nu}{2}  \| \boldsymbol{\nabla} {\bf v} \|_2^2 \, . \label{boundH}
\end{equation}
At this stage it is convenient to decompose $\textbf{v}$ into Fourier components,
\begin{equation}
\textbf{v} = \sum\limits_{\textbf{k}} \hat{\textbf{v}}_\textbf{k}(t) e^{i \textbf{k}\cdot \textbf{x}} + c.c. \, ,
\label{Fourier3D}
\end{equation}
where the wavevector $\textbf{k}$ takes the values
\begin{equation}
\textbf{k}=(k_x,k_y,k_z)=\left(n_x \frac{2\pi}{L}, n_y \frac{2\pi}{L},n_z  \frac{\pi}{H} \right), \mbox{with } {(n_x,n_y,n_z) \in \mathbb{N}\times\mathbb{N} \times\mathbb{N}^*}.
\end{equation}
The value $n_z=0$ is excluded because $\textbf{v}$ is the three-dimensional part of the velocity field and therefore it has zero vertical average.

Using this Fourier decomposition together with Parseval's equality, equation (\ref{boundH}) gives
\begin{eqnarray}
\nonumber \mathcal{H}\{\textbf{v},t\}  & \leq &c \left( \frac{ \| \boldsymbol{\nabla} \textbf{V} \|_2}{L \sqrt{H}} + \frac{\| \boldsymbol{\nabla} \textbf{V} \|_2^2}{\nu H}  \right) \| {\bf v} \|_2^2 - 2L^2 H \sum\limits_{\textbf{k}} \left(    \frac{\beta \pi^2}{k^2 H^2} + \frac{\nu}{2} k^2 \right)    | \hat{\textbf{v}}_\textbf{k}|^2 \\
 & \leq & \left[ c \left( \frac{ \| \boldsymbol{\nabla} \textbf{V} \|_2}{L \sqrt{H}} + \frac{\| \boldsymbol{\nabla} \textbf{V} \|_2^2}{\nu H}  \right) -\min_{k \in \mathbb{R}^+} \left(    \frac{\beta \pi^2}{k^2 H^2} + \frac{\nu}{2} k^2 \right) \right] \| {\bf v} \|_2^2 \, ,\label{Fourierineq}
\end{eqnarray}
which we write as
\begin{eqnarray}
\mathcal{H}\{\textbf{v},t\} & \leq &  \lambda_1(t) \| \textbf{v} \|_{2}^2 \mbox{ , with } \lambda_1(t)=  c \left( \frac{ \| \boldsymbol{\nabla} \textbf{V} \|_2}{L \sqrt{H}} + \frac{\| \boldsymbol{\nabla} \textbf{V} \|_2^2}{\nu H}  \right) - \frac{\pi}{H} \sqrt{2 \beta \nu} \, .
\end{eqnarray}

To prove the decay of the three-dimensional part of the velocity field in the long-time limit, we need the time-average of $\lambda_1$ to be negative. Using the concavity of the square-root function, we have $\left< \| \boldsymbol{\nabla} \textbf{V} \|_2 \right> \leq \sqrt{ \left< \| \boldsymbol{\nabla} \textbf{V} \|_2^2 \right>}$, hence the following upper bound on $\left< \lambda_1 \right>$,
\begin{equation}
\left< \lambda_1 \right> \leq c \left( \frac{1 }{L \sqrt{H}} \sqrt{ \left< \| \boldsymbol{\nabla} \textbf{V} \|_2^2 \right> } + \frac{1}{\nu H} \left< \| \boldsymbol{\nabla} \textbf{V} \|_2^2 \right> \right) - \frac{\pi}{H} \sqrt{2 \beta \nu} \, .
\end{equation}
If the right-hand side expression is negative, then so is $\left< \lambda_1 \right>$ and the flow becomes exactly 2D in the long-time limit. This expression is a quadratic polynomial in $\sqrt{ \left< \| \boldsymbol{\nabla} \textbf{V} \|_2^2 \right> }$. From the computation of the roots of the polynomial, we obtain that it is negative if
\begin{equation}
\left< \| \boldsymbol{\nabla} \textbf{V} \|_2^2 \right>  \leq c \sqrt{\beta} \nu^{3/2} \, ,
\label{criterionAbs}
\end{equation}
where $c$ is yet another dimensionless positive constant that is independent of the parameters of the problem.

The term $\left<  \| \boldsymbol{\nabla} \textbf{V}  \|_{2}^2 \right> $ can be bounded in terms of the viscous energy dissipation per unit mass $\epsilon$ inside the flow,
\begin{equation}
\left<  \| \boldsymbol{\nabla} \textbf{V}  \|_{2}^2 \right>  \leq \left<  \| \boldsymbol{\nabla} \textbf{u}  \|_{2}^2 \right>  = \frac{\epsilon L^2 H}{\nu} \leq \frac{U^3 L^2 H}{\ell \nu} \left( \frac{c_1}{Re} +c_2 \right) \, ,
\label{dV2}
\end{equation}
where we used the bound (\ref{DF}) on the energy dissipation of a 3D flow.
We obtain a sufficient criterion for two-dimensionalization expressed in terms of the root-mean-square velocity by demanding that the right-hand side of (\ref{criterionAbs}) be greater than the right-hand-side of (\ref{dV2}):
\begin{equation}
N \geq c\, \frac{ H^2 L^4}{ \ell^6} Re^5 \left( \frac{c_1}{Re} +c_2 \right)^2 \, .
\label{NcAbsolute}
\end{equation}

This criterion shows unambiguously that, whatever the Reynolds number is, there is a critical value of the interaction parameter above which the three-dimensional part of the velocity field decays and the flow becomes two-dimensional in the long-time limit. This two-dimensionalization occurs for three-dimensional perturbations ${\bf v}$ of arbitrary amplitude, provided this initial condition leads an rms velocity satisfying (\ref{NcAbsolute}), and we therefore call it {\it absolute two-dimensionalization}.

As a side note, we stress the difference between specifying the rms velocity $U$ and specifying the amplitude of the forcing. All the initial conditions leading to an rms velocity $U$ satisfying (\ref{NcAbsolute}) two-dimensionalize in the long-time limit. By contrast, for constant amplitude of the forcing, two different initial conditions may end up in different attractors of the system, with differing rms velocities $U_1$ and $U_2$. It may be that $U_1$ satisfies (\ref{NcAbsolute}) but $U_2$ does not.

\section{Linear two-dimensionalization\label{section4}}

In this section we come back to the full-MHD problem, with the goal of proving two-dimensionalization for finite values of $Rm$. We show that, starting from an infinitesimal three-dimensional velocity perturbation, the flow becomes two-dimensional asymptotically in time provided the interaction parameter is above some threshold value and the magnetic Reynolds number is lower than some other threshold value.

Let us decompose the velocity field into $\textbf{u}=\textbf{V}(x,y,t)+\textbf{v}(x,y,z,t)$, where $\textbf{V}(x,y,t)$ is a solution of the 2D body-forced Navier-Stokes equation, and assume that $\textbf{v}(x,y,z,t)$ and $\textbf{b}(x,y,z,t)$ are infinitesimal perturbations to this (possibly turbulent) base state. The three fields $\textbf{V}$, $\textbf{v}$ and $\textbf{b}$ are divergence-free. The linearized evolution equations for $\textbf{v}$ and $\textbf{b}$ are
\begin{eqnarray}
& & \partial_t \textbf{v} +(\textbf{v} \cdot \boldsymbol{\nabla}) \textbf{V} + (\textbf{V} \cdot \boldsymbol{\nabla}) \textbf{v} =-\boldsymbol{\nabla}{p} + \nu \Delta \textbf{v} +   \frac{1}{\rho \mu_0} B_0 \partial_z \textbf{b}\, , \label{linNS} \\
& & \partial_t \textbf{b} + (\textbf{V} \cdot \boldsymbol{\nabla}) \textbf{b} = B_0 \partial_z \textbf{v} +(\textbf{b} \cdot \boldsymbol{\nabla}) \textbf{V} +\eta \Delta  \textbf{b}\, , \label{lininduction}
\end{eqnarray}
where $p$ is the pressure perturbation.
From these equations, we compute evolution equations for the three following quadratic quantities
\begin{eqnarray}
\frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{1}{2} \|\textbf{v}\|_2^2 \right) & = &  - \int_\mathcal{D} \textbf{v} \cdot (\boldsymbol{\nabla} \textbf{V}) \cdot \textbf{v} \, \mathrm{d}^3 \textbf{x} - \nu \|\boldsymbol{\nabla} \textbf{v}\|_2^2 + \frac{B_0}{\rho \mu_0} \int_\mathcal{D} \textbf{v} \cdot \partial_z \textbf{b} \, \mathrm{d}^3 \textbf{x}  \label{eqken} \, ,\\
\frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{1}{2} \|\textbf{b}\|_2^2 \right) & = &   \int_\mathcal{D} \textbf{b} \cdot (\boldsymbol{\nabla} \textbf{V}) \cdot \textbf{b} \, \mathrm{d}^3 \textbf{x} - \eta \|\boldsymbol{\nabla} \textbf{b}\|_2^2 - \frac{B_0}{\rho \mu_0} \int_\mathcal{D} \textbf{v} \cdot \partial_z \textbf{b} \, \mathrm{d}^3 \textbf{x}  \label{eqmen} \, , \\
\frac{\mathrm{d}}{\mathrm{d}t} \left(  \int_\mathcal{D} \textbf{v} \cdot \partial_z \textbf{b} \, \mathrm{d}^3 \textbf{x} \right) & = &   \int_\mathcal{D} \partial_z\textbf{v} \cdot (\boldsymbol{\nabla} \textbf{V}) \cdot \textbf{b} \, \mathrm{d}^3 \textbf{x}  -   \int_\mathcal{D} \textbf{b} \cdot (\boldsymbol{\nabla} \textbf{V}) \cdot \partial_z \textbf{v} \, \mathrm{d}^3 \textbf{x}       \label{eqcross} \\
\nonumber & & +(\nu+ \eta)  \int_\mathcal{D} (\boldsymbol{\nabla} \partial_z\textbf{v}) \cdot (\boldsymbol{\nabla} \textbf{b}) \, \mathrm{d}^3 \textbf{x}  + B_0 \left(\frac{1}{\rho \mu_0} \|\partial_z \textbf{b}\|_2^2 - \|\partial_z \textbf{v}\|_2^2 \right) \, .
\end{eqnarray}

With the goal of determining a Lyapunov functional for the system of linearized equations, let us define
\begin{equation}
\mathcal{F}_\alpha\{\textbf{v},\textbf{b} \}=\frac{\|\textbf{v}\|_2^2 }{2}+    \frac{\|\textbf{b}\|_2^2}{2 \rho \mu_0} + \alpha   \int_\mathcal{D} \textbf{v} \cdot \partial_z \textbf{b} \, \mathrm{d}^3 \textbf{x}\, ,
\end{equation}
where $\alpha$ is a positive real number, and combine (\ref{eqken}), (\ref{eqmen}) and (\ref{eqcross}) to obtain
\begin{eqnarray}
\label{ineq1} \frac{\mathrm{d}}{\mathrm{d}t} \mathcal{F}_\alpha & = &  - \int_\mathcal{D} \textbf{v} \cdot (\boldsymbol{\nabla} \textbf{V}) \cdot \textbf{v} \, \mathrm{d}^3 \textbf{x} - \nu \|\boldsymbol{\nabla} \textbf{v}\|_2^2   + \frac{1}{\rho \mu_0}\int_\mathcal{D} \textbf{b} \cdot (\boldsymbol{\nabla} \textbf{V}) \cdot \textbf{b} \, \mathrm{d}^3 \textbf{x}  \\
\nonumber & & - \frac{\eta}{\rho \mu_0} \|\boldsymbol{\nabla} \textbf{b}\|_2^2+ \alpha \left[   \int_\mathcal{D} \partial_z\textbf{v} \cdot (\boldsymbol{\nabla} \textbf{V}) \cdot \textbf{b} \, \mathrm{d}^3 \textbf{x}  -   \int_\mathcal{D} \textbf{b} \cdot (\boldsymbol{\nabla} \textbf{V}) \cdot \partial_z \textbf{v} \, \mathrm{d}^3 \textbf{x}  \right.  \\
\nonumber & & \left.  +(\nu+ \eta)  \int_\mathcal{D} (\boldsymbol{\nabla} \partial_z\textbf{v}) \cdot (\boldsymbol{\nabla} \textbf{b}) \, \mathrm{d}^3 \textbf{x}  + B_0 \left(\frac{1}{\rho \mu_0} \|\partial_z \textbf{b}\|_2^2 - \|\partial_z \textbf{v}\|_2^2 \right)   \right] \, .
\end{eqnarray}

Because the base state is independent of the vertical and equations (\ref{linNS}) and (\ref{lininduction})  are linear, we can decompose the perturbation into Fourier modes in $z$ and study the independent evolution of a single vertical Fourier mode. In the following, we therefore consider that $(\textbf{v},\textbf{b})$ is a single vertical Fourier mode of perturbation, with vertical wavenumber $q$. Then $\|\partial_z \textbf{v}\|_2^2=q^2 \| \textbf{v}\|_2^2$ and $\|\partial_z \textbf{b}\|_2^2=q^2 \| \textbf{b}\|_2^2$. We denote as $\mathcal{F}^{(q)}_{\alpha_q}$ the functional $\mathcal{F}_\alpha$ for such a Fourier mode, where the value of $\alpha_q$ is free to depend on $q$.  Using H\"older's, the Cauchy-Schwarz and Young's inequalities, we obtain from (\ref{ineq1})
\begin{eqnarray}
\label{ineq2}   \frac{\mathrm{d}}{\mathrm{d}t}    \mathcal{F}^{(q)}_{\alpha_q}    & \leq &   \|\boldsymbol{\nabla} \textbf{V}\|_\infty \|\textbf{v}\|_2^2 - \nu \|\boldsymbol{\nabla} \textbf{v}\|_2^2 + \|\boldsymbol{\nabla} \textbf{V}\|_\infty \frac{\|\textbf{b}\|_2^2}{\rho \mu_0} -\frac{\eta}{\rho \mu_0} \|\boldsymbol{\nabla} \textbf{b}\|_2^2 \\
\nonumber & + & {\alpha_q} \left[ \|\boldsymbol{\nabla} \textbf{V}\|_\infty \left( \frac{1}{{\alpha_q}} \|\textbf{v}\|_2^2 + {\alpha_q} q^2 \|\textbf{b}\|_2^2  \right) \right.     \\
\nonumber & + & \frac{1+Pm}{\sqrt{Pm}} \sqrt{\rho \mu_0} \frac{H}{\pi} \left( \frac{\nu}{2} q^2 \|\boldsymbol{\nabla} \textbf{v}\|_2^2   + \frac{\eta \pi^2}{2 \rho \mu_0 H^2} \|\boldsymbol{\nabla} \textbf{b}\|_2^2    \right) \\
\nonumber      & + & \left. B_0 q^2\left(\frac{1}{\rho \mu_0} \| \textbf{b}\|_2^2 - \| \textbf{v}\|_2^2 \right)   \right] \, ,
\end{eqnarray}
where we used Young's inequality (together with an optimization) to get the terms in the parentheses of the second and third lines, and we introduced the magnetic Prandtl number $Pm=\nu/\eta$ to alleviate somewhat the equations.

We then decompose the Ohmic dissipation term into
\begin{equation}
-\frac{\eta}{\rho \mu_0} \|\boldsymbol{\nabla} \textbf{b}\|_2^2 \leq -\frac{\eta}{2 \rho \mu_0} \|\boldsymbol{\nabla} \textbf{b}\|_2^2 -\frac{q^2 \eta}{2\rho \mu_0} \| \textbf{b}\|_2^2 \, ,
\end{equation}
and rewrite (\ref{ineq2}) as
\begin{eqnarray}
 \frac{\mathrm{d}}{\mathrm{d}t}  \mathcal{F}^{(q)}_{\alpha_q}   & \leq & \| \textbf{v}\|_2^2 \left[ 2 \|\boldsymbol{\nabla} \textbf{V}\|_\infty  - {\alpha_q} B_0 q^2 \right] + \nu \| \boldsymbol{\nabla} \textbf{v}\|_2^2 \left[ -1 + \frac{{\alpha_q} q^2  H \sqrt{\rho \mu_0} (1+Pm)}{2 \pi \sqrt{Pm}}  \right] \label{4brackets} \\
\nonumber & + & \frac{\| \textbf{b}\|_2^2}{\rho \mu_0} \left[  \|\boldsymbol{\nabla} \textbf{V}\|_\infty (1+{\alpha_q}^2 q^2 \rho \mu_0)  + {\alpha_q} B_0 q^2  -\frac{\eta q^2}{2}\right]  \\
\nonumber &+& \eta \frac{\|\boldsymbol{\nabla} \textbf{b}\|_2^2 }{\rho \mu_0} \left[ -\frac{1}{2} + \frac{{\alpha_q} \pi^2 \sqrt{\rho \mu_0} (1+Pm)}{2 H^2 \sqrt{Pm}}    \right] \, .
\end{eqnarray}
The method is now similar to the one used in section \ref{StratAbs}: if we can find a value of $\alpha_q$ such that
\begin{itemize}
\item $\mathcal{F}^{(q)}_{\alpha_q}$ is a positive functional,
\item there exists a real function $\lambda_2(t)$ such that $\frac{\mathrm{d}}{\mathrm{d}t} \mathcal{F}^{(q)}_{\alpha_q} \leq \lambda_2(t) \mathcal{F}^{(q)}_{\alpha_q}$,
\item the time average of $\lambda_2(t)$ is strictly negative: $\left< \lambda_2(t) \right> < 0$,
\end{itemize}
then the vertical Fourier mode $q$ of the perturbation decays to zero in the long-time limit. And if we can find such an $\alpha_q$ for all Fourier modes with nonzero vertical wave number $q$ allowed in the system, then the $L_2$ norm of the $z$-dependent perturbation decays to zero in the long-time limit, and the system relaxes to the 2D solution of the Navier-Stokes equation.

To find a suitable $\alpha_q$, we first impose the constraint
\begin{equation}
0 <  \alpha_q < \frac{\pi \sqrt{Pm}}{q^2 H \sqrt{\rho \mu_0} (1+Pm)}\, . \label{C1}
\end{equation}
The reason for this constraint is made clear before equation (\ref{drop}): it ensures that the terms proportional to $\|\boldsymbol{\nabla} \textbf{v}\|_2^2$ and $\|\boldsymbol{\nabla} \textbf{b}\|_2^2$ in (\ref{4brackets}) are negative and can be discarded.

This constraint, together with $q \geq \pi/H$, Cauchy-Schwarz and Young's inequalities, leads to
\begin{eqnarray}
\left| \alpha_q  \int_\mathcal{D} \textbf{v} \cdot \partial_z \textbf{b} \, \mathrm{d}^3 \textbf{x} \right| & \leq & \alpha_q |q| \| \textbf{v}\|_2 \| \textbf{b}\|_2 \\
\nonumber & \leq & \alpha_q |q| \sqrt{\rho \mu_0} \left( \frac{\| \textbf{v}\|_2^2}{2} + \frac{\| \textbf{b}\|_2^2}{2 \rho \mu_0} \right) \\
\nonumber & \leq & \frac{\sqrt{Pm}}{1+Pm} \left( \frac{\| \textbf{v}\|_2^2}{2} + \frac{\| \textbf{b}\|_2^2}{2 \rho \mu_0} \right)\, ,
\end{eqnarray}
so that an upper and a lower bound on the functional $\mathcal{F}^{(q)}_{\alpha_q}$ are
\begin{equation}
 0 \leq \left( \frac{\| \textbf{v}\|_2^2}{2} + \frac{\| \textbf{b}\|_2^2}{2 \rho \mu_0} \right) \left( 1-\frac{\sqrt{Pm}}{1+Pm} \right)  \leq \mathcal{F}^{(q)}_{\alpha_q}  \leq \left( \frac{\| \textbf{v}\|_2^2}{2} + \frac{\| \textbf{b}\|_2^2}{2 \rho \mu_0} \right) \left( 1+\frac{\sqrt{Pm}}{1+Pm} \right)  \, ,
\end{equation}
or, since we do not care about optimizing the prefactors,
\begin{equation}
0  \leq  \frac{1}{2} \left( \frac{\| \textbf{v}\|_2^2}{2} + \frac{\| \textbf{b}\|_2^2}{2 \rho \mu_0} \right) \leq \mathcal{F}^{(q)}_{\alpha_q} \leq \frac{3}{2} \left( \frac{\| \textbf{v}\|_2^2}{2} + \frac{\| \textbf{b}\|_2^2}{2 \rho \mu_0} \right)    \, , \label{boundF}
\end{equation}
where we used the fact that $\frac{\sqrt{Pm}}{1+Pm}<\frac{1}{2}$. Inequality (\ref{boundF}) proves that $\mathcal{F}^{(q)}_{\alpha_q}$ is a positive functional.

From the constraint (\ref{C1}) and using again $q \geq \pi/H$, we show that the terms proportional to $\|\boldsymbol{\nabla} \textbf{v}\|_2^2$ and $\|\boldsymbol{\nabla} \textbf{b}\|_2^2$ in (\ref{4brackets}) are negative, and upon dropping them we get
\begin{eqnarray}
 \frac{\mathrm{d}}{\mathrm{d}t}  \mathcal{F}^{(q)}_{\alpha_q}   & \leq & \| \textbf{v}\|_2^2 \left[ 2 \|\boldsymbol{\nabla} \textbf{V}\|_\infty  - \alpha_q B_0 q^2 \right]  \label{drop}\\
 \nonumber & & +  \frac{\| \textbf{b}\|_2^2}{\rho \mu_0} \left[  \|\boldsymbol{\nabla} \textbf{V}\|_\infty (1+\alpha_q^2 q^2 \rho \mu_0)  + \alpha_q B_0 q^2  -\frac{\eta q^2}{2}\right]  \\
 \nonumber & \leq & 4 \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \left( \frac{\| \textbf{v}\|_2^2}{2} + \frac{\| \textbf{b}\|_2^2}{2 \rho \mu_0} \right) -\alpha_q B_0 q^2 \| \textbf{v}\|_2^2 + \left( \alpha_q B_0 q^2 -\frac{\eta \pi^2}{2H^2} \right) \frac{\| \textbf{b}\|_2^2}{\rho \mu_0}
\end{eqnarray}
Let us finally pick
\begin{equation}
\alpha_q=\frac{7}{B_0 q^2} \left< \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \right> \label{valalpha} \, .
\end{equation}
(remember that we consider $B_0>0$). We can do so if this value of $\alpha_q$ satisfies the constraint (\ref{C1}). Then
\begin{eqnarray}
 \frac{\mathrm{d}}{\mathrm{d}t}  \mathcal{F}^{(q)}_{\alpha_q}  & \leq & 4 \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \left( \frac{\| \textbf{v}\|_2^2}{2} + \frac{\| \textbf{b}\|_2^2}{2 \rho \mu_0} \right)-14 \left< \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \right> \frac{\| \textbf{v}\|_2^2}{2} \\
 \nonumber & & +\left(14 \left< \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \right> -\frac{\eta \pi^2}{H^2} \right) \frac{\| \textbf{b}\|_2^2}{2\rho \mu_0}\, .
\end{eqnarray}
Let us further assume that we are in the region of parameter space where
\begin{equation}
\frac{\eta \pi^2}{H^2} \geq 28 \left< \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \right>, \label{C2}
\end{equation}
so that
\begin{eqnarray}
 \frac{\mathrm{d}}{\mathrm{d}t}  \mathcal{F}^{(q)}_{\alpha_q}  & \leq & 4 \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \left( \frac{\| \textbf{v}\|_2^2}{2} + \frac{\| \textbf{b}\|_2^2}{2 \rho \mu_0} \right)-14 \left< \|\boldsymbol{\nabla}  \textbf{V}\|_\infty\right> \left( \frac{\| \textbf{v}\|_2^2}{2} + \frac{\| \textbf{b}\|_2^2}{2 \rho \mu_0} \right)  \, .
\end{eqnarray}
Using respectively the lower and upper bounds (\ref{boundF}) on $\mathcal{F}^{(q)}_{\alpha_q}$ to bound the positive and negative terms in the right-hand-side of this inequality, we finally obtain
\begin{equation}
 \frac{\mathrm{d}}{\mathrm{d}t}  \mathcal{F}^{(q)}_{\alpha_q}    \leq   \lambda_2(t) \mathcal{F}^{(q)}_{\alpha_q} \text{  , with  } \lambda_2(t)  =  \left( 8 \|\boldsymbol{\nabla}  \textbf{V}\|_\infty  -\frac{28}{3} \left< \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \right> \right)\, \label{eqlambda}
\end{equation}
Because the function $\lambda_2(t)$ has a negative time-average, the three-dimensional perturbation decays in the long-time limit.
Of course, along the way to obtain (\ref{eqlambda}) we made some assumptions that constrain the region of parameter space in which the conditions hold. Indeed, we need the chosen value (\ref{valalpha}) of $\alpha_q$ to satisfy the constraint (\ref{C1}), and the magnetic diffusivity to be large enough for (\ref{C2}) to hold, i.e., we need respectively
\begin{equation}
\frac{B_0}{\left< \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \right> H \sqrt{\rho \mu_0}} >  \frac{7(1+Pm)}{\pi \sqrt{Pm}} \text{ , and } \frac{ \left<  \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \right> H^2}{\eta} \leq \frac{\pi^2}{28} \, . \label{criterion}
\end{equation}
They involve some kind of interaction parameter and magnetic Reynolds number built with the $L_\infty$ norm of the velocity gradients. The former must be large enough and the latter small enough for magnetic damping to be efficient. Because one has little control over the infinite norm of the velocity gradients, we wish to express these constraints using the rms velocity only.

\section{The large Reynolds number regime\label{section5}}

We focus on the large-$Re$ regime and therefore consider $Re \geq 1$. Then, for a solution ${\bf V}(x,y,t)$ of the body-forced two-dimensional Navier-Stokes equation, we can bound the time-averaged $L_\infty$ norm of the velocity gradients $\left< \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \right>$ in terms of the enstrophy dissipation rate. This task is performed in appendix \ref{Appendix2}, with the following result,
\begin{eqnarray}
\left< \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \right> & \leq &  c_5 \sqrt{ \frac{\left< \| \boldsymbol{\nabla} \omega \|_2^2 \right>}{H}} \left( 1+ \ln Re + \ln \frac{L}{\ell}  \right) \, ,
 \label{bndLinfty}
\end{eqnarray}
where the $H$ at the denominator arises from the fact that $\| \dots \|_2$ is $L_2$ norm in 3D, and $c_5$ depends on the shape of the forcing only. The bounds (\ref{2Dbound}) and (\ref{2DboundKolmogorov}) on $\left< \| \boldsymbol{\nabla} \omega \|_2^2 \right>$ then allow to bound $\left< \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \right>$ in terms of the rms velocity $U$.

\subsection{Time-independent forcing}

For a time-independent forcing, let us combine the upper bound on the vorticity gradients (\ref{2Dbound}) with equation (\ref{bndLinfty}) to get
\begin{eqnarray}
\left< \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \right>  & \leq & c_6 \frac{U L}{\ell^2}  \sqrt{Re} \,  \left( 1+ \ln Re + \ln \frac{L}{\ell}  \right) \, ,
\end{eqnarray}
which we substitute into the inequalities (\ref{criterion}) to get alternate sufficient criteria for two-dimensionalization, which now involve only the rms velocity of the base flow:
\begin{eqnarray}
N  & \geq &  c_{7} \frac{H^2 L^2}{\ell^4} Re^2 \left( 1 +  \ln Re +  \ln  \frac{L}{\ell}   \right)^2 \, , \label{condition1}\\
\text{and } Rm & \leq &  \frac{\ell^3}{H^2 L} \, \frac{c_{8}}{\sqrt{Re} \left( 1 +  \ln Re +  \ln  \frac{L}{\ell}   \right) } \, , \label{condition2}
\end{eqnarray}
where we also assumed $Re \geq 1$ and $Pm \leq 1$ along the way, and $c_{7}$ and $c_{8}$ are positive numbers that depend on the forcing shape only. Forgetting about the logarithmic terms, the first condition requires the magnetic field to be strong enough for the interaction parameter to be larger than $Re^2$. Note that the prefactor is proportional to $H^2$, so that the constraint is more easily met in shallow domains. The second condition is to have a low enough magnetic Reynolds number, lower than roughly the inverse square-root of the Reynolds number. Once again, because the prefactor is inversely proportional to $H^2$ this constraint is more easily satisfied in shallow domains. Condition (\ref{condition2}) can be recast in terms of the kinetic Reynolds number using $Rm=Pm \, Re$, which gives
\begin{equation}
Re \leq  \frac{\ell^2}{(H^2 L)^{\frac{2}{3}}}  \, \left [\frac{c_{8}}{Pm \left( 1 +  \ln Re +  \ln  \frac{L}{\ell}   \right)} \right]^{\frac{2}{3}} \, .
\end{equation}
Neglecting the log term on the right-hand-side and assuming that the geometrical prefactor is of order unity, the maximum Reynolds number for which we can prove two-dimensionalization is roughly $Pm^{-2/3}$. For a liquid metal $Pm \simeq 10^{-6}$, so this is typically $10^4$. Once again, in shallow domains (small $H/\ell$) two-dimensionalization can be proven up to higher $Re$.

\subsection{Kolmogorov forcing}

The tighter upper bound (\ref{2DboundKolmogorov}) on the time-averaged vorticity gradients holds if the forcing function is of single-mode type, that is if $\Delta \textbf{f}$ is proportional to $\textbf{f}$.
For such forcing we combine (\ref{2DboundKolmogorov}) with equation (\ref{bndLinfty}) to get a bound on the velocity gradients in terms of the rms velocity
\begin{eqnarray}
\left< \|\boldsymbol{\nabla}  \textbf{V}\|_\infty \right>  & \leq & c_{9} \frac{U L}{\ell^2} \, \left( 1 + \ln Re + \ln \frac{L}{\ell}  \right) \, ,\end{eqnarray}
which we insert into (\ref{criterion}) to obtain criteria for two-dimensionalization in terms of the rms velocity only:
\begin{eqnarray}
N  & \geq & c_{10} \frac{H^2 L^2}{\ell^4} Re \left( 1 + \ln Re + \ln \frac{L}{\ell}  \right)^2 \, , \label{condition1Kolm}\\
\text{and } Rm & \leq & \frac{\ell^3}{H^2 L} \, \frac{c_{11}}{ \left( 1 + \ln Re + \ln \frac{L}{\ell}  \right) } \, , \label{condition2Kolm}
\end{eqnarray}
where we also assumed $Re \geq 1$ and $Pm \leq 1$ along the way, and $c_{10}$ to $c_{11}$ are numbers which depend on the shape of the forcing only. We can see that the assumption of single-mode forcing allows to prove two-dimensionalization in a larger region of parameter space. Indeed, forgetting about the logarithmic terms, conditions (\ref{condition1Kolm}) and (\ref{condition2Kolm}) indicate that if the interaction parameter is larger than the Reynolds number and the magnetic Reynolds number is smaller than a constant, then the flow remains two-dimensional. Once again, the geometrical prefactors in (\ref{condition1Kolm}) and (\ref{condition2Kolm}) are such that the sufficient conditions for two-dimensionalization are more easily met in shallow domains. Condition (\ref{condition2Kolm}) can be recast in terms of the kinetic Reynolds number only as
\begin{equation}
Re \leq \frac{1}{Pm} \, \frac{\ell^3}{H^2 L} \,  \frac{c_{11}}{ \left( 1 + \ln Re + \ln \frac{L}{\ell}  \right) }  \, . \label{condition2KolmRe}
\end{equation}
For Kolmogorov forcing, we can therefore prove two-dimensionalization up to a Reynolds number of order $Pm^{-1}$. For a liquid metal, this is typically $10^6$. This maximum value of $Re$ for which we can prove two-dimensionalization is larger in shallow domains because of the prefactor $\ell^3/(H^2 L)$.

\section{Discussion\label{section6}}

\begin{table}
\begin{center}
\begin{tabular*}{0.95\textwidth}{@{\extracolsep{\fill}}  c | c | c  }
System & Region of parameter space & Property\\
 & (asymptotic in $Re$) & \\
\hline
Full MHD, time-independent forcing & $\displaystyle N \geq  \text{const} \times \frac{H^2 L^2}{\ell^4} Re^2 [\ln(Re)]^2$ & L2D \\
 & $\displaystyle Rm \leq \text{const} \times \frac{\ell^3}{H^2 L \sqrt{Re} \ln(Re)} $ & \\
 \hline
 Full MHD, single-mode, time-independent forcing & $\displaystyle N \geq  \text{const} \times \frac{H^2 L^2}{\ell^4} Re [\ln(Re)]^2$ & L2D \\
 & $\displaystyle Rm \leq \text{const} \times \frac{\ell^3}{H^2 L \ln(Re)} $ & \\
 \hline
Quasi-static MHD, time-independent forcing & $\displaystyle N \geq  \text{const} \times \frac{H^2 L^4}{\ell^6} Re^5 $ & A2D \\
 \hline
\end{tabular*}
\caption{Regions of parameter space where we proved two-dimensionalization. The threshold values are written in the limit of large Reynolds number. L2D stands for ``linear two-dimensionalization" and A2D for ``absolute two-dimensonalization". \label{table1}}
\end{center}
\end{table}

\begin{figure}
\begin{center}
\includegraphics[width=120 mm]{fig2a.eps}
\includegraphics[width=120 mm]{fig2b.eps}
\end{center}
\caption{A sketch of the bounds (\ref{NcAbsolute}), (\ref{condition1Kolm}) and (\ref{condition2KolmRe}), in parameter space, for single-mode forcing, $\ell=L/2$, and several values of the aspect ratio $H/L$. For simplicity, the dimensionless constants appearing in these expressions have been taken equal to unity. \textbf{Top:} Results obtained in the quasi-static limit. Above the linear two-dimensionalization line (blue, solid line), we have proven that the flow has a stable 2D attractor. However, this attractor is stable for small 3D perturbations only, and fully 3D attractors of the system may exist as well. Above the absolute two-dimensionalization line (red dashed), the flow has only 2D attractors. \textbf{Bottom:} Linear two-dimensionalization for the full-MHD system, for magnetic Prandtl number $Pm=10^{-6}$ (blue solid) and $Pm=10^{-5}$ (red dashed): on the top left part of each curve, we have proven that the system has a purely two-dimensional attractor.\label{scalingQS}}
\end{figure}

In this study we considered a flow driven by a horizontal body-force and subject to a strong vertical magnetic field. We determined regions of parameter space where the two-dimensional solutions of the Navier-Stokes equation are stable with respect to three-dimensional perturbations. Such two-dimensionalization occurs for strong enough interaction parameter $N$ and low enough magnetic Reynolds number $Rm$. We first focused on the $Rm \ll 1$ asymptotic limit, the so-called quasi-static limit, to prove absolute two-dimensionalization: above some $Re$-dependent value of the interaction parameter, the attractors of the flow are all 2D, i.e., the flow becomes 2D in the long-time limit, regardless of the initial condition. We then came back to the full-MHD problem, for which we derived a value $N_c(Re)$ of the interaction parameter and a value $Rm_c(Re)$ of the magnetic Reynolds number such that, if $N\geq N_c$ and $Rm \leq Rm_c$, the 2D flow is stable with respect to infinitesimal vertically-dependent perturbations: we called this property linear two-dimensionalization. Of course, the results proven using the full-MHD equations are also valid in the quasi-static limit $Rm \to 0$: then only $N\geq N_c$ is required for linear two-dimensionalization. Alternatively, one can directly prove linear two-dimensionalization for quasi-static MHD: bound the triple velocity product in (\ref{defH}) by $\| \boldsymbol{\nabla} {\bf V} \|_\infty \| {\bf v} \|_2^2$, where ${\bf V}$ is now a solution of the 2D Navier-Stokes equation. The dissipative terms are bounded as in (\ref{Fourierineq}), hence $\mathcal{H}\{\textbf{v},t\} \leq ( \| \boldsymbol{\nabla} {\bf V} \|_\infty -\sqrt{2 \beta \nu}\pi/H)  \| {\bf v} \|_2^2$. Asking for the time-average of the parenthesis to be negative leads to the condition $\frac{\sqrt{\beta \nu}}{H \left< \| \boldsymbol{\nabla} {\bf V} \|_\infty \right>} \geq \frac{1}{\sqrt{2} \pi}$ for linear two-dimensionalization. Conditions expressed in terms of the rms velocity $U$ are obtained using (\ref{bndLinfty}) followed by (\ref{2Dbound}) or (\ref{2DboundKolmogorov}): this gives the same criteria as (\ref{condition1}) and (\ref{condition1Kolm}), albeit with different numerical constants.

The different regions of two-dimensionalization are summarized in Table \ref{table1}, and sketched in figure \ref{scalingQS} in the case of single-mode forcing. These results have implications for the existence of an energy dissipation anomaly in this system.
Indeed, as can be seen in top panel of figure \ref{scalingQS}, depending on how one takes the double limit $(Re\to \infty, N \to \infty)$, the quasi-static system may or may not exhibit a dissipation anomaly. If we take this limit with $N>Re^5$, then in the long-time limit the flow settles to a purely 2D attractor, with vanishing Ohmic dissipation, and a time-averaged viscous dissipation that follows a laminar scaling, proportional to $Re^{-1}$: there is no dissipation anomaly (on a purely mathematical basis, our proof of absolute two-dimensionalization shows that the 3D part of the flow decays in the $L_2$ sense only, and one could argue that the 3D velocity gradients can remain large. However, physical intuition suggests that the 3D velocity gradients decay when the $L_2$ norm of the 3D velocity decays).

In the region $N>Re (\ln Re)^2$, we know that the system has a 2D attractor that is stable with respect to infinitesimal perturbations. If the initial condition is close enough to being 2D, the system will end up in this purely 2D state, with no energy dissipation anomaly. However, this attractor is not necessarily the only one of the dynamics: if the initial condition is strongly 3D, the flow may end up in a fully-3D attractor, and this 3D state may exhibit a dissipation anomaly.

The limiting values of the interaction parameter derived in the present study are rather conservative: the best criterion for two-dimensionalization we produced is roughly $N \geq Re (\ln Re)^2$, when several examples of 2D flows become stable as soon as $N$ is larger than some $Re$-independent value (see for instance the study of the strained elliptical vortex by \citet{Thess}). It could be that the computed threshold values of $N$ are not sharp, or that the 2D turbulent flows we considered are much more unstable to 3D perturbations than the smooth ideal 2D flows considered by previous authors.

We now wish to comment on the shape of the domain: we see that all the threshold values of $N$  in Table \ref{table1} are proportional to the squared aspect ratio $H^2/L^2$ of the domain, and the threshold $Rm$ are inversely proportional to it: the shallower the domain, the larger the region of parameter space in which we can prove two-dimensionalization, as can be seen in figure \ref{scalingQS}. That is because shallow domains prohibit the long-vertical-wavelength perturbations that are poorly affected by magnetic damping

Further comments are needed regarding the boundary conditions considered in this study.
We first stress the fact that the results of this study carry over to 3D periodic domains (albeit with different dimensionless numerical constants), which are more fashionable among numericists. As far as laboratory experiments are concerned, the simple magnetic boundary conditions in (\ref{BC}) can be relevant: they correspond to boundaries with a much higher magnetic permeability than the fluid. As an example, electromagnets often contain iron pieces to channel the magnetic field. If liquid metal is placed in the gap of such an electromagnet, between two iron pieces, the boundaries have very large magnetic permeability provided the ferromagnetic material is not saturated: the magnetic boundary conditions then approach those in (\ref{BC}). Recently, such large-magnetic-permeability boundaries have attracted attention in the dynamo community, because in some geometries they lower the threshold for dynamo action \citep{Gissinger, Gallet2012, Gallet2013, Herault}. However, these are not the most common boundaries: many experiments  and astrophysical situations correspond to non-magnetic and insulating boundaries. These alternate boundary conditions do not affect the results obtained here in the quasi-static approximation, because equation (\ref{NSzeroRm}) still applies. Indeed, both the high-magnetic-permeability and the insulating boundary conditions impose a vanishing normal component of the current density at the boundary, and therefore Neumann boundary conditions for the inverse Laplacian in equation (\ref{NSzeroRm}). As a result, both absolute and linear two-dimensionalization can be proven for insulating boundaries in the quasi-static limit, with the same criteria as for high-magnetic-permeability boundaries.
 However, the proof of linear two-dimensionalization for the full-MHD equations does not carry over to non-magnetic boundary conditions, although we believe similar results probably hold.

Finally, although the stress-free or 3D-periodic boundary conditions on the velocity field might have some relevance in an astrophysical context, they are a crude approximation to liquid metal experiments, where the fluid flows inside a rigid container. Such no-slip boundaries cannot be taken into account in the present analysis, because they prevent the existence of a purely 2D state. Instead, for strong enough magnetic field the flow settles in a quasi-2D state, with vertically invariant flow in the bulk and sharp Hartmann layers at the horizontal boundaries \citep{Sommeria, Klein, Potherat2014}. These layers can induce additional instabilities \citep{Lingwood, Moresco}, and dissipation through Hartmann friction, which we avoid by considering stress-free or periodic boundary conditions.

Some of this work was performed at the Geophysical Fluid Dynamics Program at Woods Hole Oceanographic Institution, supported by the US National Science Foundation (NSF) Awards OCE-0824636 and OCE-1332750 and the Office of Naval Research.  And some of this work was completed at the NSF Institute for Pure \& Applied Mathematics at UCLA.  This research was also supported in part by NSF Award PHY-1205219 (CRD) and by the junior grant TURBA from Labex PALM ANR-10-LABX-0039 (BG).

redpajama_set_name: RedPajamaArXiv


text:
US & World/
Policy/
Qualcomm must license patents to competing chipmakers, court rules
Ruling is part of larger battle with FTC over anti-competitive practices
By Jacob Kastrenakes / @jake_k
Nov 6, 2018, 9:59 PM UTC |
Qualcomm was dealt a major loss in one of its many ongoing legal battles this afternoon, with a federal court ruling that the company must license its modem patents to competing chipmakers, potentially weakening its stranglehold on the market.
The ruling came out of a Federal Trade Commission lawsuit against Qualcomm, which was filed near the start of 2017. The crux of the lawsuit — whether Qualcomm used anti-competitive practices to maintain a monopoly over smartphone modems — isn't being ruled on here. But the court did hone in on a single question at issue: whether Qualcomm has to license standard essential patents to competitors.
The court ruled it "unambiguous" that Qualcomm was wrong
In this case, the court ruled it does. Qualcomm agreed to two separate policies that said it would offer select patents on a non-discriminatory basis. Those patents were essential to wireless standards — and only accepted into the standards because of Qualcomm's agreement to license to everyone. After looking at the contracts, the court said it was "unambiguous" that Qualcomm was wrong here. Qualcomm did not immediately respond to a request for comment.
If Qualcomm were allowed to keep its standard essential patents to itself, the court wrote, it would enable the company "to achieve a monopoly in the modem chip market and limit competing implementations of those components." That phrasing may be concerning for Qualcomm, as whether it used anti-competitive practices to maintain a monopoly is at issue in the rest of the lawsuit.
The ruling means that Qualcomm has to license patents necessary for building a smartphone modem to competing companies, like Intel. Until now, Qualcomm has only offered those licenses to companies that directly manufacture smartphones, and it seems that Qualcomm only did that when it was directly selling chips to them.
This could lead to more or better modems from other companies
That's meant that a company like Intel, which badly wants to compete with Qualcomm in this market, has had to work around Qualcomm's patents in order to sell modems of its own. And it means that a company like Apple or Samsung, that wants to sell a ton of smartphones, has largely had to rely on Qualcomm's chips in order to do so.
That's bad news for Qualcomm, but good news for the rest of the industry. Should the ruling hold, it could enable more companies to build modems or for those modems to be more competitive than they are today. Intel, for instance, already makes competing modems, but they've never been as fast as Qualcomm's.
What doesn't seem to be changing, however, is how much Qualcomm can charge for those patents. The FTC has also accused Qualcomm of charging too-high fees for its patents, despite these same agreements requiring it to only impose "reasonable" fees. Apple is suing Qualcomm over the very same issue, but courts have yet to rule on this.
Qualcomm faces similar legal battles around the globe. Apple is suing it in multiple locations over onerous licensing fees, while Qualcomm has already been fined in South Korea, Taiwan, the European Union, and China for issues related to anti-competitive licensing practices.
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Apr 16, 2019, 11:51 PM UTC


text:
Swiss Gigolo Gets 6 Years
Updated Apr. 25, 2017 4:02PM ET / Published Mar. 09, 2009 2:43AM ET
Raymond Chandler couldn't have written a better plot: The Times of London reports that Helg Sgarbi, 43, will go on trial today for allegedly blackmailing Germany's richest woman, BMW heiress Susanne Klatten, 46, out of millions of euros. Reportedly the grandson of a forced laborer, Sgarbi's possible motive may have been revenge against Klatten's grandfather, who advised Hitler on economic policy. The lawyer Sgarbi chose, Egon Geiss, 77, made his name defending Nazis and may try to stir up stories from the Nazi past. Supposedly, Sgarbi first bilked Klatten out of $8.8 million by telling her that a US Mafia don was threatening to kill him after accidentally running over the man's daughter in Florida. Klatten delivered, but when she tried to end the affair after admitting it to her husband, Sgarbi sent her compromising pictures of the two of them and asked for $62 million.
Read it at The Times of London


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"If you only see one cabaret show this year, see Lady Rizo… Soulful, uproarious, slightly scary and utterly irresistible. ★★★★★"
- Time Out
Since awarding her the Time Out and Soho Theatre Cabaret Award at the Edinburgh Festival Fringe in 2012, Soho Theatre have brought Lady Rizo to their stage for three sell-out seasons. Now, New York City's prized cabaret superstar explores how fecundity and parenthood fit with the revelry of showbiz. Backed by a choice London trio, Rizo rips apart carefully chosen pop songs and her own stirring originals. An entertainment machine even at four months post-partum, her soulful vocals and titillating humour will leave you breathless in her gloved palm…just don't inhale the baby powder. Running Time: 1 hour 30 minutes. Recommended for ages 16 and over. Valid ID may be required.
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