Input: What is the dynamical behavior of the anisotropic order parameter following a quench to the critical point?

Context:
\section*{Dynamical Behaviour of $O$ in Lattice Gases}

The dynamical behaviour of the anisotropic order parameter  $m$ [see Eq.~\eqref{eq:def-m} in the Letter] following a quench to the critical point is well described by
the Gaussian theory for all the three lattice gas models studied, $i.e.,$ driven lattice gas with either constant (IDLG) or random (RDLG) infinite drive and equilibrium lattice gas (LG). In other words, in the short-time regime, $m \sim t^{1/2}$ [see Eq. \eqref{eq:mt}] and the Binder cumulant $g$ of the lowest transverse mode [defined in Eq. \eqref{eq:binder}] is zero in this regime. The alternative order parameter $O,$ however, distinguishes between the driven (IDLG, RDLG) and the equilibrium (LG) lattice gases.

In order to understand  this, we first write the phenomenological scaling form for $O$,  analogous to Eq. \eqref{eq:scalingass} in  the Letter,
\begin{eqnarray}
O (t, L_{\parallel} ; S_\Delta) = L_{\parallel}^{-\beta/[\nu(1+\Delta)]} \tilde f_O (t/L_{\parallel}^{z/(1+\Delta)} ; S_\Delta).\quad
\label{eq:Oscalingass}
\end{eqnarray}
We already remarked that, in the LG, this scaling form is not compatible with the prediction $O \sim t^{1/8}  L_{\parallel}^{-1/2}$ of the Gaussian theory.  However, following Ref. \cite{AS2002}, it can be argued that, at short times, the only dependence of $O$ on the system size $L_{\parallel}$ is of the form $O \sim L_\parallel^{-1/2}$ which is very well confirmed by numerical simulations. Accordingly,  the generic behaviour of $O$ can be assumed to be
\begin{eqnarray}
O \sim t^{\alpha} L_\parallel^{-1/2}, \label{eq:O}
\end{eqnarray}
where $\alpha$ is a phenomenological exponent to be determined. This, along with Eq. \eqref{eq:Oscalingass}, implies $\tilde f_O(x) \sim x^{\alpha}.$ Comparing the finite-size behaviour in Eq.~\eqref{eq:O} with Eq.~\eqref{eq:Oscalingass} one actually infers,
\begin{eqnarray}
\alpha &=& \frac{1+ \Delta -2 \beta/\nu}{2 \, (4- \eta)}. \label{eq:alpha}
\end{eqnarray}
This equation, together with the hyperscaling relation $\Delta - 2 \beta/\nu= - \eta$ in two spatial dimensions, shows that the prediction $\alpha = 1/8$ of the Gaussian theory [see Eq. \eqref{eq:Ot}] can be obtained only when $\eta=0,$ which is the case for the IDLG (exactly) and the RDLG (approximately) but not for the LG.

On the other hand,  Eq.~\eqref{eq:alpha} predicts $\alpha = 1/10$ upon substituting the values of the critical exponents corresponding to the Ising  universality class (LG). This is consistent with the numerical simulation results presented in the main text, see Fig. \ref{fig:ising}(b) therein.

\begin{figure}[th]
\vspace*{0.2 cm}
 \centering
 \includegraphics[width=10 cm]{./compare_binder.pdf}

\caption{Comparison between the temporal evolution of the Binder cumulants $g$ corresponding to the $12^{th}$ transverse mode, $i.e.,$ with $n_\perp =12,$ in the LG (lowest curve), IDLG and RDLG (two upper curves) on a $32 \times 32$ lattice. \label{fig:b}}
 \label{fig:binder}
\end{figure}

The emergence of this new value $1/10$ of the exponent $\alpha$ must be traced back to the non-Gaussian nature of higher fluctuating modes in the LG. In fact, even though the lowest mode behaves identically in all the three models we considered,  characterized by the same behaviour of $m$, higher modes show a significant difference in the non-driven case.

To illustrate this, we measured the Binder cumulants of higher modes which is defined  analogously to Eq.~(11), using transverse modes other than the first, i.e., with $\mu=\tilde \sigma(0,2 \pi n_\bot/L_\bot)$ and $n_\bot>1.$
 Figure \ref{fig:b} compares the same for all the three lattice gases for the mode with $n_\perp =12$ on a $32 \times 32$ lattice. Clearly, the curve corresponding to the LG (lowest, blue) departs from Gaussian behaviour $g=0$ (in practice, $e.g.,$ $|g| \lesssim 0.005,$ corresponding to the shaded gray area) much earlier than it does for the IDLG  or RDLG (two upper curves, red and green respectively).

Accordingly, the different dynamical behaviour of $O$, which involves a sum over all modes, can be attributed to the non-Gaussian nature of the higher modes in the LG.
Such a departure is not entirely surprising. In fact, for higher modes, mesoscopic descriptions such as the ones in Eqs. \eqref{eq:L-DLG} or \eqref{eq:g_evol} are not expected to hold, while the anisotropy at the microscopic level could be the mechanism leading to the Gaussianity of higher modes in the driven models.

Answer: It is well described by the Gaussian theory.


Input: When did Born resign as chairperson of the CFTC?

Context:
Brooksley Elizabeth Born (born August 27, 1940) is an American attorney and former public official who, from August 26, 1996, to June 1, 1999, was chair of the Commodity Futures Trading Commission (CFTC), the federal agency which oversees the U.S. futures and commodity options markets. During her tenure on the CFTC, Born lobbied Congress and the President to give the CFTC oversight of off-exchange markets for derivatives, in addition to its role with respect to exchange-traded derivatives, but her warnings were ignored or dismissed, and her calls for reform resisted by other regulators.<ref name="nytimes">Goodman, Peter S. The Reckoning - Taking Hard New Look at a Greenspan Legacy, The New York Times, October 9, 2008.</ref> Born resigned as chairperson on June 1, 1999, shortly after Congress passed legislation prohibiting her agency from regulating derivatives.

In 2009, Born received the John F. Kennedy Profiles in Courage Award, along with Sheila Bair of the Federal Deposit Insurance Corporation, in recognition of the "political courage she demonstrated in sounding early warnings about conditions that contributed" to the 2007-08 financial crisis.

Early life and education
Born graduated from Abraham Lincoln High School (San Francisco, California) at the age of 16. She then attended Stanford University, where she majored in English and was graduated with the class of 1961.  She initially wanted to become a doctor, but a guidance counsellor at Stanford advised her against medicine, so she majored in English literature instead.

She then attended Stanford Law School, one of only seven women in her class.  She was the first female student ever to be named president of the Stanford Law Review. She received the "Outstanding Senior" award and graduated as valedictorian of the class of 1964.

Legal career
Immediately after law school Born was selected as a law clerk to judge Henry Edgerton of the U.S. Court of Appeals for the District of Columbia Circuit. It was during this time that she met her first husband, Jacob C. Landau, who was a journalist covering the Federal courts at the time. Following her clerkship, she became an associate at the Washington, D.C.-based international law firm of Arnold & Porter. Born was attracted to Arnold & Porter because it was one of the few major law firms to have a woman partner at that time, Carolyn Agger, who was the head of the tax practice. Born took a two-year leave of absence from Arnold & Porter to accompany her first husband to Boston, where he had received a fellowship. During that time she worked as a research assistant to law professor Alan Dershowitz.

Born's early career at Arnold & Porter focused on international trade law, in which she represented a number of Swiss industries and the government of Switzerland. She developed a practice representing clients in numerous complex litigation and arbitration cases involving financial market transactions. Among her high-profile cases was the matter of the Hunt Brothers attempt to corner the market in silver in the 1970s. She made partner at Arnold & Porter, after moving to a three-day schedule to help raise her second child, and eventually rose to be the head of the firm's derivatives practice.

Born was among the first female attorneys to systematically address inequities regarding how the laws treated women. Born and another female lawyer, Marna Tucker, taught what is considered to have been the first "Women and the Law" course at Catholic University’s Columbus School of Law. The class exclusively concerned prejudicial treatment of women under the laws of the United States, past and present.  Born and Tucker were surprised to discover that there was no textbook on the issue at the time.  Born is also one of the co-founders of the National Women's Law Center. Born also helped rewrite the American Bar Association rules to make it possible for more women and minorities to sit on federal bench.

During her long legal career, and into her retirement, Born did much pro bono and other types of volunteer work. She was active in the American Bar Association, the largest professional organization of lawyers in the United States.  Initially Born was named a member of the governing council of the ABA's Individual Rights Section, eventually becoming chairperson.  Born and Tucker founded the ABA Women's Caucus, the first organization of female lawyers in the ABA.  She held several other senior positions in the ABA, including being named the first woman member of the ABA's Standing Committee on the Federal Judiciary.  As a member of the Judiciary Committee, Born provided testimony and opinion on persons nominated for federal judgeships. In 1980 she was named chair of the committee.  As chair of the committee, Born was invited to address the U.S. Congress regarding the nomination of Judge Sandra Day O'Connor to the U.S. Supreme Court.

In 1993, Born's name was floated as a possible candidate for Attorney General of the United States, but Janet Reno was nominated.

In July 2009, Nancy Pelosi appointed Brooksley Born as a commissioner to the Financial Crisis Inquiry Commission (FCIC).

Born and the OTC derivatives market
Born was appointed to the CFTC on April 15, 1994, by President Bill Clinton. Due to litigation against Bankers Trust Company by Procter and Gamble and other corporate clients, Born and her team at the CFTC sought comments on the regulation of over-the-counter derivatives, a first step in the process of writing CFTC regulations to supplement the existing regulations of the Federal Reserve System,  the Options Clearing Corporation, and the National Association of Insurance Commissioners. Born was particularly concerned about swaps, financial instruments that are traded over the counter between banks, insurance companies or other funds or companies, and thus have no transparency except to the two counterparties and the counterparties' regulators, if any.  CFTC regulation was strenuously opposed by Federal Reserve chairman Alan Greenspan, and by Treasury Secretaries Robert Rubin and Lawrence Summers. On May 7, 1998, former SEC Chairman Arthur Levitt joined Rubin and Greenspan in objecting to the issuance of the CFTC's concept release. Their response dismissed Born's analysis and focused on the hypothetical possibility that CFTC regulation of swaps and other OTC derivative instruments could create a "legal uncertainty" regarding such financial instruments,  hypothetically reducing the value of the instruments. They argued that the imposition of regulatory costs would "stifle financial innovation" and encourage financial capital to transfer its  transactions offshore. The disagreement between Born and the Executive Office's top economic policy advisors has been described not only as a classic Washington turf war, but also a war of ideologies,  insofar as it is possible to argue that Born's actions were consistent with Keynesian and neoclassical economics while Greenspan, Rubin, Levitt, and Summers consistently espoused neoliberal, and neoconservative policies.

In 1998, a trillion-dollar hedge fund called Long Term Capital Management (LTCM) was near collapse.  Using mathematical models to calculate debt risk, LTCM used derivatives to leverage $5 billion into more than $1 trillion, doing business with fifteen of Wall Street's largest financial institutions.  The derivative transactions were not regulated, nor were investors able to evaluate LTCM's exposures.  Born stated, "I thought that LTCM was exactly what I had been worried about".  In the last weekend of September 1998, the President's working group was told that the entire American economy hung in the balance.  After intervention by the Federal Reserve, the crisis was averted.  In congressional hearings into the crisis, Greenspan acknowledged that language had been introduced into an agriculture bill that would prevent CFTC from regulating the derivatives which were at the center of the crisis that threatened the US economy.  U.S. Representative Maurice Hinchey (D-NY) asked "How many more failures do you think we'd have to have before some regulation in this area might be appropriate?"  In response, Greenspan brushed aside the substance of Born's warnings with the simple assertion that "the degree of supervision of regulation of the over-the-counter derivatives market is quite adequate to maintain a degree of stability in the system". Born's warning was that there wasn't any regulation of them.  Born's chief of staff, Michael Greenberger summed up Greenspan's position this way: "Greenspan didn't believe that fraud was something that needed to be enforced, and he assumed she probably did. And of course, she did."  Under heavy pressure from the financial lobby, legislation prohibiting regulation of derivatives by Born's agency was passed by the Congress.  Born resigned on June 1, 1999.

The derivatives market continued to grow yearly throughout both terms of George W. Bush's administration. On September 15, 2008, the bankruptcy of Lehman Brothers forced a broad recognition of a financial crisis in both the US and world capital markets.  As Lehman Brothers' failure temporarily reduced financial capital's confidence, a number of newspaper articles and television programs suggested that the failure's possible causes included the conflict between the CFTC and the other regulators.Faiola, Anthony, Nakashima, Ellen and Drew, Jill. The Crash: Risk and Regulation - What Went Wrong, The Washington Post, October 15, 2008.

Born declined to publicly comment on the unfolding 2008 crisis until March 2009, when she said: "The market grew so enormously, with so little oversight and regulation, that it made the financial crisis much deeper and more pervasive than it otherwise would have been." She also lamented the influence of Wall Street lobbyists on the process and the refusal of regulators to discuss even modest reforms.

An October 2009 Frontline documentary titled "The Warning"  described Born's thwarted efforts to regulate and bring transparency to the derivatives market, and the continuing opposition thereto. The program concluded with an excerpted interview with Born sounding another warning: "I think we will have continuing danger from these markets and that we will have repeats of the financial crisis -- may differ in details but there will be significant financial downturns and disasters attributed to this regulatory gap, over and over, until we learn from experience."

In 2009 Born, along with Sheila Bair of the FDIC, was awarded the John F. Kennedy Profiles in Courage Award in recognition of the "political courage she demonstrated in sounding early warnings about conditions that contributed" to the 2007-08 financial crisis.  According to Caroline Kennedy, "Brooksley Born recognized that the financial security of all Americans was being put at risk by the greed, negligence and opposition of  powerful and well connected interests.... The catastrophic financial events of recent months have  proved them [Born and Sheila Bair] right."  One member of the President's working group had a change of heart about Brooksley Born.  SEC Chairman Arthur Levitt stated "I've come to know her as one of the most capable, dedicated, intelligent and committed public servants that I have ever come to know", adding that "I could have done much better. I could have made a difference" in response to her warnings.

In 2010, a documentary film Inside Job further alleged that derivatives regulation was ineffective from the Clinton administration on. Along with fellow whistleblower, former IMF Chief Economist Raghuram Rajan, who was also scorned by the economic establishment, Brooksley Born was cited as one of the authorities arguing that financial derivatives increase economic risk.

 Personal life
Born is married to Alexander E. Bennett (also retired from Arnold & Porter).  She has five adult children - two from a previous marriage to Jacob Landau and three stepchildren. Notably, Born was named a partner at Arnold & Porter while working part-time so she could raise her two young children.  When both of her children were school-age, Born returned to practice full-time.

References

External links
Attorney profile at Arnold & Porter
Brooksley Born (2009 Winner) of the Profiles in Courage Award, with acceptance speech transcript and NECN video

Profile at MarketsWiki
Speeches and statements
"Testimony Of Brooksley Born Chairperson of the CFTC Concerning The Over-The-Counter Derivatives Market", before the House Committee On Banking And Financial Services, July 24, 1998.
"The Lessons of Long Term Capital Management L.P.", Remarks of Brooksley Born, Chairperson of the CFTC, Chicago-Kent-IIT Commodities Law Institute, Chicago, Illinois, October 15, 1998.
 Interview: Brooksley Born for "PBS Frontline: The Warning", PBS, (streaming VIDEO 1 hour), October 20, 2009.
Articles
Manuel Roig-Franzia. "Credit Crisis Cassandra:Brooksley Born's Unheeded Warning Is a Rueful Echo 10 Years On", The Washington Post, May 26, 2009
 Taibbi, Matt. "The Great American Bubble Machine", Rolling Stone'', July 9–23, 2009

1940 births
American women lawyers
Arnold & Porter people
Clinton administration personnel
Columbus School of Law faculty
Commodity Futures Trading Commission personnel
Heads of United States federal agencies
Lawyers from San Francisco
Living people
Stanford Law School alumni
21st-century American women
Stanford University alumni

Answer: June 1, 1999.


Input: When did the 2017 general election be held?

Context:
Sir Simon William English  (born 30 December 1961) is a New Zealand former National Party politician who served as the 39th prime minister of New Zealand from 2016 to 2017. He had previously served as the 17th deputy prime minister of New Zealand and minister of finance from 2008 to 2016 under John Key and the Fifth National Government.

A farmer and public servant before entering politics, English was elected to the New Zealand Parliament in  as the National Party's candidate in the Wallace electorate. He was elevated to Cabinet in 1996 and in 1999 was made minister of finance, although he served for less than a year due to his party's loss at the 1999 general election. In October 2001, English replaced Jenny Shipley as the leader of the National Party (and consequently as Leader of the Opposition). He led the party to its worst defeat at the 2002 general election, and as a consequence, in October 2003 he was replaced as leader by Don Brash.

In November 2006, after Brash's resignation, English became deputy leader under John Key. After National's victory at the 2008 general election, he became deputy prime minister and was also made minister of finance for the second time. Under English's direction New Zealand's economy maintained steady growth during National's three terms of government. He became a list-only MP after stepping down as an electorate MP at the 2014 general election.

John Key resigned as leader of the National Party and prime minister in December 2016. English won the resulting leadership election unopposed and was sworn in as prime minister on 12 December 2016. His tenure was only ten months, and included a three-month election campaign. In the 2017 general election, National won the largest number of seats but fell short of a majority. The parties holding the balance of power declined to support the existing government, and English was subsequently replaced as prime minister by Jacinda Ardern, leader of the Labour Party. English initially continued on as Leader of the Opposition, but resigned as leader of the National Party on 27 February 2018 and left parliament two weeks later.

Early life
English was born on 30 December 1961 at Lumsden Maternity Centre in Lumsden. He is the eleventh of twelve children of Mervyn English and Norah (née O'Brien) English. His parents purchased Rosedale, a mixed sheep and cropping farm in Dipton, Southland from Mervyn's uncle, Vincent English, a bachelor, in 1944. English was born in the maternity unit at Lumsden.

English attended St Thomas's School in Winton, then boarded at St. Patrick's College in Upper Hutt, where he became head boy. He played in the first XV of the school's rugby team. English went on to study commerce at the University of Otago, where he was a resident at Selwyn College, and then completed an honours degree in English literature at Victoria University of Wellington.

After finishing his studies, English returned to Dipton and farmed for a few years. From 1987 to 1989, he worked in Wellington as a policy analyst for the New Zealand Treasury, at a time when the free market policies favoured by Labour's finance minister Roger Douglas (known collectively as "Rogernomics") were being implemented.

English joined the National Party in 1980, while at Victoria University. He served for a period as chairman of the Southland branch of the Young Nationals, and became a member of the Wallace electorate committee. After moving to Wellington, he served for periods on the Island Bay and Miramar electorate committees, respectively.

Fourth National Government (1990–1999)

At the 1990 general election, English stood as the National candidate in Wallace, replacing the retiring Derek Angus, and was elected with a large majority. He would hold this seat, renamed Clutha-Southland in 1996, until 2014. He and three other newly elected National MPs (Tony Ryall, Nick Smith, and Roger Sowry) were soon identified as rising stars in New Zealand politics, and at various points were dubbed the "brat pack", the "gang of four", and the "young Turks". In his first term in parliament, English chaired a select committee into social services. He was made a parliamentary under-secretary in 1993, serving under the Minister of Health.

First period in cabinet (1996–1999)
In early 1996, English was elevated to cabinet by Prime Minister Jim Bolger, becoming the Minister for Crown Health Enterprises and Associate Minister of Education (to Wyatt Creech). He was 34 at the time, becoming the cabinet's youngest member. After the 1996 general election, the National Party was forced into a coalition with New Zealand First to retain government. In the resulting cabinet reshuffle, English emerged as Minister of Health. However, as a condition of the coalition agreement, NZ First's Neil Kirton (a first-term MP) was made Associate Minister of Health, effectively becoming English's deputy. This arrangement was described in the press as a "shotgun marriage", and there were frequent differences of opinion between the two ministers. After their relationship became unworkable, Kirton was sacked from the role in August 1997, with the agreement of NZ First leader Winston Peters.

As Minister of Health, English was responsible for continuing the reforms to the public health system that National had begun after the 1990 general election. The reforms were unpopular, and health was perceived as one of the government's weaknesses, with the health portfolio consequently being viewed as a challenge. English believed that the unpopularity of the reforms was in part due to a failure in messaging, and encouraged his National colleagues to avoid bureaucratic and money-focused language (such as references to "balance sheets" and "user charges") and instead talk about the improvements to services the government's reforms would bring. He also rejected the idea that public hospitals could be run as commercial enterprises, a view which some of his colleagues had previously promoted.

By early 1997, as dissatisfaction with Bolger's leadership began to grow, English was being touted as a potential successor, along with Jenny Shipley and Doug Graham. His age (35) was viewed as the main impediment to a successful leadership run. National's leadership troubles were resolved in December 1997, when Bolger resigned and Shipley was elected to the leadership unopposed. English had been a supporter of Bolger as leader, but Shipley reappointed him Minister of Health in her new cabinet.

English was promoted to Minister of Finance in a reshuffle in January 1999, a position which was at the time subordinate to the Treasurer, Bill Birch. After a few months, the pair switched positions as part of Birch's transition to retirement, with English assuming the senior portfolio. In early interviews, he emphasised his wish to be seen as a pragmatist rather than an ideologue, and said that the initiatives of some of his predecessors (Roger Douglas's "Rogernomics" and Ruth Richardson's "Ruthanasia") had focused on "fruitless, theoretical debates" when "people just want to see problems solved".

Opposition (1999–2008)

After the National Party lost the 1999 election to Helen Clark's Labour Party, English continued on in the shadow cabinet as National's spokesperson for finance. He was elected deputy leader of the party in February 2001, following the resignation of Wyatt Creech, with Gerry Brownlee being his unsuccessful opponent.

Leader of the Opposition
In October 2001, after months of speculation, Jenny Shipley resigned as leader of the National Party after being told she no longer had the support of the party caucus. English was elected as her replacement unopposed (with Roger Sowry as his deputy), and consequently became Leader of the Opposition. However, he did not openly organise against Shipley, and according to The Southland Times "there was almost an element of 'aw, shucks, I'll do it then' about Mr English's ascension".

Aged 39 when he was elected, English became the second-youngest leader in the National Party's history, after Jim McLay (who was 38 when elected in 1984). He also became only the third Southlander to lead a major New Zealand political party, after Joseph Ward and Adam Hamilton. However, English failed to improve the party's performance. In the 2002 election, National suffered its worst electoral defeat ever, gaining barely more than twenty percent of the vote. English described it as "the worst day of my political life". Both party insiders and the general public were split as to how much to blame him for the loss, but most of the party believed that English would be able to rebuild National's support.

By late 2003, however, National's performance in opinion polls remained poor. The party had briefly increased its popularity in the year following the election, but by October its support had fallen to levels only slightly better than what it achieved in the last ballot. English also appeared in a boxing match for a charity against entertainer Ted Clarke. This did not boost his polling or that of the National party either, with suggestions that it devalued his image as a serious politician. Don Brash, former governor of the Reserve Bank and a relative newcomer to politics, began to build up support to replace English. On 28 October, Brash gained sufficient backing in Caucus to defeat English in a leadership contest.

Shadow cabinet roles and deputy leader
On 2 November 2003, when Brash changed responsibilities for certain MPs, English became National's spokesman for education, ranked at fifth place in the party's parliamentary hierarchy. He remained in parliament after the 2005 election. In his new shadow education portfolio, English performed strongly, and remained a party favourite despite his election defeat as leader in 2002, eventually being returned to the finance portfolio in August 2004 as deputy spokesman (while still retaining responsibility for education).

In November 2006, Brash resigned as leader. English was considered as a potential replacement leader (running against John Key) or deputy leader (against incumbent Gerry Brownlee) in the ensuing leadership election. However, a contest was avoided when the MPs agreed a Key/English ticket would run unopposed in a display of party unity. English took over the deputy leadership and the finance portfolio in the Key shadow cabinet.

Fifth National Government (2008–2017)

Deputy Prime Minister and Minister of Finance (2008–2016)

At the 2008 election, English was re-elected by his electorate, winning by a margin of about 15,500 votes. He became Deputy Prime Minister of New Zealand and Minister of Finance in the fifth National Government, being sworn into office on 19 November 2008 and continued to serve in those roles until becoming Prime Minister on 12 December 2014. He was also made Minister of Infrastructure in National's first term of government and Minister responsible for Housing New Zealand Corporation and minister responsible for the New Zealand flag consideration process in its third.

He was comfortably re-elected in Clutha-Southland in the 2011 election but opted to run as a party-list candidate in 2014.

The pairing of John Key as leader of the National Party and English as his deputy has been compared to that of Bob Hawke and Paul Keating (in Australia) and Tony Blair and Gordon Brown (in the UK).

English acceded to the role of Finance Minister in the continuing wake of the financial crisis. In response to New Zealand's rising debt, English made budget deficit-reduction his main priority. His first budget outlined three focuses in New Zealand's financial recovery: "improving the business environment and removing roadblocks to growth; investment in productive infrastructure; and improving the way government works". One of his first acts was creating the National Infrastructure Unit, charged with formulating a plan for infrastructure projects and investments. He commissioned a government-wide spending review, with an aim to reducing government expenditure—with the exceptions of a two-year stimulus package and long-term increases on infrastructure spending.

In April 2011, the Opposition criticised English for suggesting that New Zealand businesses could use New Zealand's low wages to help it compete with Australia. The National Government campaigned for re-election in 2011 on its economic record. The Government boasted growth for five consecutive quarters up to mid-2010, totalling 1.6% of real GDP.

Strong growth resulted in a surplus of $473 million for the 2015/16 financial year, projected to rise to $8.5 billion by 2020/21. In his 2016 Economic and Fiscal Update address, English stated that reducing debt and tackling the costs of the 2016 Kaikōura earthquake were higher priorities than reducing rates of tax.

Allowances issue
In 2009, the media, including TVNZ and TV3 revealed that English was receiving about NZ$900 a week as part of a living allowance for ministers, to live in his own NZ$1.2 million Wellington home. At the time, English also received $276,200 in his annual salary as Deputy Prime Minister. It was also revealed other ministers with homes in the capital city were also claiming accommodation allowances. On 3 August 2009, Prime Minister John Key started a review of the housing allowances claimed by cabinet ministers. English subsequently paid back $12,000 and only claimed about $24,000 a year in living allowances. The Auditor-General's office said in September 2009 that they were making "preliminary enquiries" into parliamentary housing expenses in response to a letter of complaint from Progressive party leader Jim Anderton. Two days later English stated that he would no longer take up any housing allowance and had paid back all the allowance he had received since the November 2008 election.

Prime Minister (2016–2017)

John Key resigned on 12 December, and endorsed English as his successor in the resulting leadership election. Following the drop-out of both Judith Collins and Jonathan Coleman from the leadership election, English was sworn in as the 39th Prime Minister of New Zealand on 12 December 2016.

English appointed his first cabinet on 18 December. In a reshuffle, he appointed Steven Joyce to succeed him as Finance Minister, while most ministerial portfolios remained the same.

In February 2017, English did not attend Waitangi Day commemorations at the historic treaty grounds, reportedly in response to the Ngāpuhi iwi's decision to stop the Prime Minister from speaking at the marae. Ngāpuhi have protested the Government's negotiation of the Trans Pacific Partnership Agreement (TPPA), which the iwi believe infringes upon Māori sovereignty, and thus does not adhere to the Treaty of Waitangi. English had been invited to attend in an official capacity; his non-attendance was criticised by a Ngāpuhi elder and Opposition leader Andrew Little.

In his first overseas trip as Prime Minister, English travelled to Europe to discuss trade ties, including a prospective New Zealand–European Union free trade agreement. He first travelled to London on 13 January 2017 to meet British Prime Minister Theresa May. Discussing trade relations, English said the two nations were "natural partners" and would "continue to forge ties" after the UK's withdrawal from the EU. He also arranged to meet with London Mayor Sadiq Khan, Belgian Prime Minister Charles Michel and German Chancellor Angela Merkel.  In a meeting with Merkel, English received crucial backing from Germany for a trade deal with the EU. On 16 January, English stated that his government would continue to promote TPPA, despite the United States' decision to withdraw from the agreement. He explained that Southeast Asian countries would now be treated as a priority in negotiations—he also asserted that the United States was ceding influence to China by its rejection of the trade pact.

At a press conference at the Beehive on 1 February 2017, English announced that the 2017 general election would be held on 23 September. The Prime Minister later confirmed that his party would approach ACT, United Future and the Māori Party if confidence and supply agreements were required to form a government following the election. In his second cabinet reshuffle on 24 April, English appointed Gerry Brownlee as his new Foreign Affairs Minister; he also promoted Nikki Kaye to the portfolio of Education Minister, and moved Mark Mitchell into the cabinet to become Defence Minister. The reshuffle was perceived as an election preparation.

On 13 February 2017, English welcomed Australian Prime Minister Malcolm Turnbull to Wellington. The two leaders reaffirmed their shared trade agenda, and discussed changes to the Australian citizenship pathway which will affect permanent residents originating from New Zealand.

On 19 June, it was reported that Todd Barclay, who succeeded English as MP for Clutha-Southland, had clandestinely recorded one of his employee's conversations the previous year, and that John Key's leaders' budget was used to pay a confidential settlement after the employee resigned. English admitted that he had been aware of the illegal recording and the settlement, and thus implicated in the scandal.

During the 2017 National campaign launch, English introduced a $379 million social investment package including digital learning academies for high school students, more resources for mathematics, and boosting support for teaching second languages in schools, and maintaining National Standards in the school curriculum. Prime Minister English also sought to defend National's financial management and economic track record and claimed that the opposition Labour Party would raise taxes. Early opinion polling had forecast a poor showing in the election for the Labour Party, but in early August 37-year-old Jacinda Ardern took over as Labour leader and seemingly energised younger voters.

At the 2017 general election, National won the largest share of the party vote (44.4%) and the largest number of seats (56) in the House Representatives. However, National lacked enough seats to govern alone due to two of the party's support partners, the Māori Party and United Future, losing their parliamentary seats. In response, English stated that the party would be entering into talks to form a coalition with New Zealand First. Following talks with the two largest parties, New Zealand First entered a coalition arrangement with the Labour Party. English was succeeded as prime minister by Jacinda Ardern on 26 October.

Opposition (2017–2018)

Leader of the Opposition
English was re-elected as National Party leader on 24 October 2017. At the time of his re-election, English announced his intention to stay on as leader until the next general election. On 13 February 2018, however, he stood down as National Party leader due to personal reasons, and instructed the party to put into motion the processes to elect a new leader. He also retired from Parliament. English's resignation followed weeks of speculation that he would step aside for a new leader. On 27 February, he was succeeded as party leader by Simon Bridges as the result of the leadership election held that day.

Post-premiership
In 2018, English joined the board of Australian conglomerate, Wesfarmers. English serves in Chairmanships of Mount Cook Alpine Salmon, Impact Lab Ltd and Manawanui Support Ltd. He is also a director of The Instillery, Centre for Independent Studies and The Todd Corporation Limited, and is a member of the Impact Advisory Group of Macquarie Infrastructure and Real Assets.

Political and social views

English is regarded as more socially conservative than his predecessor, John Key. He has stated his opposition to voluntary euthanasia and physician-assisted suicide, same-sex civil unions, and the decriminalisation of prostitution. As Prime Minister he opposed any "liberalisation" of abortion law.

In 2004, English voted against a bill to establish civil unions for both same-sex and opposite-sex couples. In 2005, he voted for the Marriage (Gender Clarification) Amendment Bill, which would have amended the Marriage Act to define marriage as only between a man and a woman. English voted against the Marriage (Definition of Marriage) Amendment Bill, a bill that legalised same-sex marriage in New Zealand. However, in December 2016 he stated, "I'd probably vote differently now on the gay marriage issue. I don't think that gay marriage is a threat to anyone else's marriage".

In 2009, English voted against the Misuse of Drugs (Medicinal Cannabis) Amendment Bill, a bill aimed at amending the Misuse of Drugs Act so that cannabis could be used for medical purposes.

Personal life
English met his future wife, Mary Scanlon, at university. She was studying medicine at the time, and became a general practitioner. Both her parents were immigrants, her father being Samoan and her mother Italian, born on the island of Stromboli. They have six children: a daughter and five sons.

English is a practising Roman Catholic, but has stated that he considers his religious beliefs personal and thus separate from politics.

In June 2002, English took part in TV3's Fight For Life, a celebrity boxing fundraiser to raise money for the Yellow Ribbon anti-youth-suicide campaign, influenced by the death of a teenage nephew in 1997. He lost a split decision to former university colleague Ted Clarke.

Honours
In the 2018 Queen's Birthday Honours, English was appointed a Knight Companion of the New Zealand Order of Merit, for services of over 27 years to the State.

Answer: 23 September.


Input: What is the relationship between the maximum velocity and the amplitude of the blob or depletion?

Context:
\section{Model equations} \label{sec:equations}

In drift-fluid models the continuity equation
\begin{align}
 \frac{\partial n}{\partial t} + \nabla\cdot\left( n \vec u_E  \right) &= 0 \label{eq:generala}
\end{align}
describes the dynamics of the electron density $n$. Here
$\vec u_E := (\hat{\vec b} \times \nabla \phi)/B$ gives the electric drift
velocity in a magnetic field $\vec B := B \hat{\vec b}$ and an electric
potential $\phi$. We neglect contributions of the diamagnetic drift~\cite{Kube2016}.

Equation~\eqref{eq:generala} is closed by invoking quasineutrality, i.e. the divergence of the ion polarization,
the electron diamagnetic and the gravitational drift currents must vanish
\begin{align}
  \nabla\cdot\left( \frac{n}{\Omega} \left( \frac{\partial}{\partial t}
  + \vec u_E \cdot\nabla  \right)\frac{\nabla_\perp \phi}{B}  + n\vec u_d - n\vec u_g\right) &=0
  .

  \label{eq:generalb}
\end{align}
Here we denote
$\nabla_\perp\phi/B := - \hat{\vec b} \times \vec u_E$,
the electron diamagnetic drift
$\vec u_d := - T_e(\hat{\vec b} \times\nabla n ) /enB$
with the electron temperature $T_e$,
the ion gravitational drift velocity
$\vec u_g := m_i \hat{\vec b} \times \vec g /B$
with ion mass $m_i$, and the ion gyro-frequency
$\Omega := eB/m_i$.

Combining Eq.~\eqref{eq:generalb} with Eq.~\eqref{eq:generala} yields
\begin{align}
 \frac{\partial \rho}{\partial t} + \nabla\cdot\left( \rho\vec u_E \right) + \nabla \cdot\left( n(\vec u_\psi + \vec u_d + \vec u_g) \right) &= 0\label{eq:vorticity}
\end{align}
with the polarization charge density
$\rho = \nabla\cdot( n\nabla_\perp \phi / \Omega B)$
and
$\vec u_\psi := \hat{\vec b}\times \nabla\psi /B$
with
$\psi:= m_i\vec u_E^2 /2e$.
We exploit this form of Eq.~\eqref{eq:generalb} in our numerical simulations.

Equations~\eqref{eq:generala} and \eqref{eq:generalb} respectively \eqref{eq:vorticity} have several invariants.
First, in Eq.~\eqref{eq:generala} the relative particle number
$M(t) := \int \mathrm{dA}\, (n-n_0)$ is conserved over time
$\d M(t)/\d t = 0$.
Furthermore, we integrate
$( T_e(1+\ln n) -T_e \ln B)\partial_t n$
as well as
$-e\phi \partial_t\rho - (m_i\vec u_E^2/2+gm_ix - T_e\ln B)\partial_t n$
over the domain to get, disregarding boundary contributions,
\begin{align}
  \frac{\d}{\d t}\left[T_eS(t) + H(t) \right] = 0, \label{eq:energya}\\
    \frac{\d}{\d t} \left[ E(t) - G(t) - H(t)\right] =  0,
    \label{eq:energyb}
\end{align}
where we define
the entropy
$S(t):=\int \mathrm{dA}\, [n\ln(n/n_0) - (n-n_0)]$,
the kinetic energy
$E(t):=m_i \int \mathrm{dA}\, n\vec u_E^2/2$
and the potential energies
$G(t) := m_i g\int \mathrm{dA}\, x(n-n_0)$
and
$H(t) := T_e\int \mathrm{dA}\, (n-n_0) \ln (B^{-1})$.
Note that $n\ln( n/n_0) - n + n_0 \approx (n-n_0)^2/2$ for $|(n-n_0)/n_0| \ll 1$ and $S(t)$ thus reduces to the
local entropy form in Reference~\cite{Kube2016}.

We now set up a gravitational field $\vec g = g\hat x$ and a constant homogeneous background
magnetic field $\vec B = B_0 \hat z$ in a Cartesian coordinate system.
Then the divergences of the electric and gravitational drift velocities $\nabla\cdot\vec u_E$ and $\nabla\cdot\vec u_g$
and the diamagnetic current $\nabla\cdot(n\vec u_d)$ vanish, which makes the
flow incompressible. Furthermore, the magnetic potential energy vanishes $H(t) = 0$.

In a second system we model the inhomogeneous magnetic field present in tokamaks as
$\vec B := B_0 (1+ x/R_0)^{-1}\hat z$ and neglect the gravitational drift $\vec u_g = 0$.
Then, the potential energy $G(t) = 0$.
Note that
$H(t) = m_i \ensuremath{C_\mathrm{s}}^2/R_0\int\mathrm{dA}\, x(n-n_0) +\mathcal O(R_0^{-2}) $
reduces to $G(t)$ with the effective gravity $g_\text{eff}:= \ensuremath{C_\mathrm{s}}^2/R_0$ with $\ensuremath{C_\mathrm{s}}^2 := T_e/m_i$.
For the rest of this letter we treat $g$ and $g_\text{eff}$ as well as $G(t)$ and $H(t)$ on the same footing.
The magnetic field inhomogeneity thus entails compressible flows, which is
the only difference to the model describing dynamics in a homogeneous magnetic field introduced above.
Since both $S(t)\geq 0$ and $E(t)\geq 0$ we further derive from Eq.~\eqref{eq:energya} and Eq.~\eqref{eq:energyb} that the kinetic energy
is bounded by $E(t) \leq T_eS(t) + E(t) = T_e S(0)$; a feature absent from the gravitational system with
incompressible flows, where $S(t) = S(0)$.

We now show that the invariants Eqs.~\eqref{eq:energya} and \eqref{eq:energyb} present restrictions on the velocity and
acceleration of plasma blobs.
First, we define the blobs' center of mass (COM) via $X(t):= \int\mathrm{dA}\, x(n-n_0)/M$ and
its COM velocity as $V(t):=\d X(t)/\d t$.
The latter is proportional to the total radial particle flux~\cite{Garcia_Bian_Fundamensky_POP_2006, Held2016a}.
We assume
that $n>n_0$ and $(n-n_0)^2/2 \leq [ n\ln (n/n_0) - (n-n_0)]n $ to show for both systems
\begin{align}
  (MV)^2 &= \left( \int \mathrm{dA}\, n{\phi_y}/{B} \right)^2
  = \left( \int \mathrm{dA}\, (n-n_0){\phi_y}/{B} \right)^2\nonumber\\

&\leq 2 \left( \int \mathrm{dA}\, \left[n\ln (n/n_0) -(n-n_0)\right]^{1/2}\sqrt{n}{\phi_y}/{B}\right)^2\nonumber\\

  &\leq 4 S(0) E(t)/m_i

  \label{eq:inequality}
\end{align}
Here we use the Cauchy-Schwartz inequality and
$\phi_y:=\partial\phi/\partial y$.
Note that although we derive the inequality Eq.~\eqref{eq:inequality} only for amplitudes $\triangle n >0$  we assume that the results also hold for depletions. This is justified by our numerical results later in this letter.
If we initialize our density field with a seeded blob of radius $\ell$ and amplitude $\triangle n$ as
\begin{align}
  n(\vec x, 0) &= n_0 + \triangle n \exp\left( -\frac{\vec x^2}{2\ell^2} \right), \label{eq:inita}

\end{align}
and
$\phi(\vec x, 0 ) = 0$,
we immediately have $M := M(0) = 2\pi \ell^2 \triangle n$, $E(0) = G(0) = 0$ and
$S(0) = 2\pi \ell^2 f(\triangle n)$, where $f(\triangle n)$ captures the amplitude dependence of
the integral for $S(0)$.

The acceleration for both incompressible and compressible flows can be estimated
by assuming a linear acceleration $V=A_0t$ and $X=A_0t^2/2$~\cite{Held2016a} and using
$E(t) = G(t) = m_igMX(t)$ in Eq.~\eqref{eq:inequality}
\begin{align}
  \frac{A_0}{g} =  \mathcal Q\frac{2S(0)}{M} \approx \frac{\mathcal Q}{2} \frac{\triangle n }{n_0+2\triangle n/9}.
  \label{eq:acceleration}
\end{align}
Here, we use the Pad\'e approximation of order $(1/1)$ of $2S(0)/M $
and define a model parameter $\mathcal Q$ with $0<\mathcal Q\leq1$ to be determined by numerical simulations.
Note that the Pad\'e approximation is a better approximation than a simple
truncated Taylor expansion especially for large relative amplitudes of order unity.
Eq.~\eqref{eq:acceleration} predicts that $A_0/g\sim \triangle n/n_0$ for small
amplitudes $|\triangle n/n_0| < 1$ and $A_0 \sim g $ for very large amplitudes $\triangle n /n_0 \gg 1$,
which confirms the predictions in~\cite{Pecseli2016} and reproduces the limits discussed in~\cite{Angus2014}.

As pointed out earlier for compressible flows Eq.~\eqref{eq:inequality} can be further estimated
\begin{align}
  (MV)^2  \leq 4 T_eS(0)^2/m_i.
  \label{}
\end{align}
We therefore have a restriction on the maximum COM velocity for compressible flows, which is absent for incompressible flows
\begin{align}
  \frac{\max |V|}{\ensuremath{C_\mathrm{s}}} = {\mathcal Q}\frac{2S(0)}{M} \approx \frac{\mathcal Q}{2} \frac{|\triangle n| }{n_0+2/9 \triangle n } \approx \frac{\mathcal Q}{2} \frac{|\triangle n|}{n_0}.
  \label{eq:linear}
\end{align}
For $|\triangle n /n_0|< 1$ Eq.~\eqref{eq:linear} reduces to the linear scaling derived in~\cite{Kube2016}.
Finally, a scale analysis of Eq.~\eqref{eq:vorticity} shows that~\cite{Ott1978, Garcia2005, Held2016a}
\begin{align}
  \frac{\max |V|}{\ensuremath{C_\mathrm{s}}} = \mathcal R \left( \frac{\ell}{R_0}\frac{|\triangle n|}{n_0} \right)^{1/2}.
  \label{eq:sqrt}
\end{align}
This equation predicts a square root dependence of the center of mass velocity
on amplitude and size.

We now propose a simple phenomenological model that captures the essential dynamics
of blobs and depletions in the previously stated systems. More specifically
the model reproduces the acceleration Eq.~\eqref{eq:acceleration} with and without
Boussinesq approximation, the square root scaling for the COM velocity
Eq.~\eqref{eq:sqrt} for incompressible flows as well as the relation between the
square root scaling Eq.~\eqref{eq:sqrt} and the linear scaling
Eq.~\eqref{eq:linear} for compressible flows.
The basic idea is that the COM of blobs behaves like
the one of an infinitely long plasma column immersed in an ambient plasma.
The dynamics of this column reduces to the one of a two-dimensional ball.
This idea is similar to the analytical ``top hat'' density solution for
blob dynamics recently studied in~\cite{Pecseli2016}.
The ball is subject to buoyancy as well as linear and nonlinear friction
\begin{align}
  M_{\text{i}} \frac{d V}{d t} = (M_{\text{g}} - M_\text{p}) g - c_1 V  - \mathrm{sgn}(V ) \frac{1}{2}c_2 V^2.
  \label{eq:ball}
\end{align}
The gravity $g$ has a positive sign in the coordinate system; sgn$(f)$ is the sign function.
The first term on the right hand side is the buoyancy, where
$M_{\text{g}} := \pi \ell^2 (n_0 + \mathcal Q \triangle n/2)$
is the gravitational mass of the ball with radius $\ell$ and
$M_\mathrm{p} := n_0 \pi \ell^2 $
is the mass of the displaced ambient plasma.
Note that if $\triangle n<0$ the ball represents a depletion and the buoyancy term has a negative sign, i.e. the depletion will rise.
We introduce an inertial mass
$M_{\text{i}} := \pi\ell^2 (n_0 +2\triangle n/9)$
different from the gravitational mass $M_{\text{g}}$ in order to
recover the initial acceleration in Eq.~\eqref{eq:acceleration}.
We interpret the parameters $\mathcal Q$ and $2/9$ as geometrical factors
that capture the difference of the actual blob form from the idealized
``top hat'' solution.
Also note that the Boussinesq approximation appears in the model as a neglect of inertia, $M_{\text{i}} = \pi\ell^2n_0$.

The second term is the linear friction term with coefficient $c_1(\ell)$, which
depends on the size of the ball.
If we disregard the nonlinear friction, $c_2=0$, Eq.~\eqref{eq:ball} directly yields a
maximum velocity $c_1V^*=\pi \ell^2 n g \mathcal Q\triangle n/2$.
From our previous considerations $\max V/\ensuremath{C_\mathrm{s}}=\mathcal Q \triangle n /2n_0$, we thus identify
\begin{align}
  c_1 = \pi\ell^2 n_0 g/\ensuremath{C_\mathrm{s}}.
  \label{}
\end{align}
The linear friction coefficient thus depends on the gravity and the size of the
ball.

The last term in \eqref{eq:ball} is the nonlinear friction. The sign of the force depends on whether
the ball rises or falls in the ambient plasma.
If we disregard linear friction $c_1=0$, we have the maximum velocity
$V^*= \sigma(\triangle n)\sqrt{\pi \ell^2|\triangle n| g\mathcal Q/c_2}$,
which must equal
$\max V= \sigma(\triangle n) \mathcal R \sqrt{g \ell |\triangle n/n_0|}$
and thus
\begin{align}
  c_2 = {\mathcal Q\pi n_0\ell }/{\mathcal R^2}.
  \label{}
\end{align}
Inserting $c_1$ and $c_2$ into Eq.~\eqref{eq:ball}
we can derive the maximum absolute velocity in the form
\begin{align}
  \frac{\max |V|}{\ensuremath{C_\mathrm{s}}} =
        \left(\frac{\mathcal R^2}{\mathcal Q}\right) \frac{\ell}{R_0} \left(
        \left({1+\left( \frac{\mathcal Q}{\mathcal R} \right)^{2} \frac{|\triangle n|/n_0 }{\ell/R_0}}\right)^{1/2}-1 \right)
  \label{eq:vmax_theo}
\end{align}
and thus have a concise expression for $\max |V|$ that captures both the linear
scaling \eqref{eq:linear} as well as the square root scaling \eqref{eq:sqrt}.
With Eq.~\eqref{eq:acceleration} and Eq.~\eqref{eq:sqrt} respectively Eq.~\eqref{eq:vmax_theo} we
finally arrive at an analytical expression for the time at which the maximum velocity is reached via
$t_{\max V} \sim \max V/A_0$. Its inverse $\gamma:=t_{\max V}^{-1}$ gives the
global interchange growth rate, for which an empirical expression was
presented in Reference~\cite{Held2016a}.

We use the open source library FELTOR
to simulate
Eqs.~\eqref{eq:generala} and \eqref{eq:vorticity} with and without
drift compression.
For numerical stabilty we added small diffusive terms on the right hand
sides of the equations.
The discontinuous Galerkin methods employ three polynomial coefficients and a minimum of $N_x=N_y=768$ grid cells. The box size is $50\ell$ in order to mitigate
influences of the finite box size on the blob dynamics.
Moreover, we used the invariants in Eqs. \eqref{eq:energya} and \eqref{eq:energyb} as consistency tests to verify the code and repeated simulations
also in a gyrofluid model.
No differences to the results presented here were found.
Initial perturbations on the particle density field are given by Eq.~\eqref{eq:inita},
where the perturbation amplitude $\triangle n/n_0$ was chosen between $10^{-3}$ and $20$ for blobs and $-10^0$ and $ -10^{-3}$ for depletions.
Due to computational reasons we show results only for $\triangle n/n_0\leq 20$.

For compressible flows we consider two different cases $\ell/R_0 = 10^{-2}$ and
$\ell /R_0 = 10^{-3}$.
 For incompressible flows Eq.~\eqref{eq:generala} and \eqref{eq:vorticity}
 can be normalized such that the blob radius is absent from the equations~\cite{Ott1978, Kube2012}.
 The simulations of incompressible flows can thus be used for both sizes.
The numerical code as well as input parameters and output data can be found
in the supplemental dataset to this contribution~\cite{Data2017}.

\begin{figure}[htb]
    \includegraphics[width=\columnwidth]{com_blobs}
    \caption{
      The maximum radial COM velocities of blobs for compressible and incompressible flows are shown.
      The continuous lines show Eq.~\eqref{eq:vmax_theo} while the
      dashed line shows the square root scaling Eq.~\eqref{eq:sqrt} with
      $\mathcal Q = 0.32$ and $\mathcal R=0.85$.
    }
    \label{fig:com_blobs}
\end{figure}
In Fig.~\ref{fig:com_blobs} we plot the maximum COM velocity for blobs
with and without drift compression.
For incompressible flows blobs follow the square root scaling almost
perfectly. Only at very large amplitudes velocities are slightly below
the predicted values.
For small amplitudes we observe that the compressible blobs follow
a linear scaling. When the amplitudes increase there is a transition to the
square root scaling at around $\triangle n/n_0 \simeq 0.5$ for
$\ell/R_0=10^{-2}$ and $\triangle n/n_0 \simeq 0.05$ for $\ell/R_0=10^{-3}$, which is consistent with Eq.~\eqref{eq:vmax_theo} and Reference~\cite{Kube2016}.
In the transition regions the simulated velocities are slightly larger than the predicted ones from Eq.~\eqref{eq:vmax_theo}.
Beyond these amplitudes
the velocities of compressible and incompressible blobs align.

\begin{figure}[htb]
    \includegraphics[width=\columnwidth]{com_holes}
    \caption{
      The maximum radial COM velocities of depletions for compressible and incompressible flows are shown.
      The continuous lines show Eq.~\eqref{eq:vmax_theo} while the
      dashed line shows the square root scaling Eq.~\eqref{eq:sqrt} with
      $\mathcal Q = 0.32$ and $\mathcal R=0.85$.
      Note that small amplitudes are on the right and amplitudes close to unity are on the left side.
  }
    \label{fig:com_depletions}
\end{figure}
In Fig.~\ref{fig:com_depletions} we show the maximum radial COM velocity
for depletions instead of blobs.
For relative amplitudes below $|\triangle n|/n_0 \simeq 0.5$ (right of unity in the plot) the velocities
coincide with the corresponding blob velocities in Fig.~\ref{fig:com_blobs}.
 For amplitudes larger than $|\triangle n|/n_0\simeq 0.5$ the
velocities follow the square root scaling.
We observe that for plasma depletions beyond $90$ percent the velocities
in both systems reach a constant value that is very well predicted by the
square root scaling.

\begin{figure}[htb]
    \includegraphics[width=\columnwidth]{acc_blobs}
    \caption{
      Average acceleration of blobs for compressible and incompressible flows are shown.
      The continuous line shows the acceleration in Eq.~\eqref{eq:acceleration}
      with $\mathcal Q=0.32$
      while the dashed line is a linear reference line, which corresponds to the Boussinesq approximation.
  }
    \label{fig:acc_blobs}
\end{figure}
In Fig.~\ref{fig:acc_blobs} we show the average acceleration of blobs
for compressible and incompressible flows computed
by dividing the maximum velocity $\max V$ by the time
to reach this velocity $t_{\max V}$.
We compare the simulation results
to the theoretical predictions Eq.~\eqref{eq:acceleration} of our model with and without inertia.
The results of the compressible and incompressible systems coincide and fit very
well to our theoretical values.
For amplitudes larger than unity the acceleration deviates significantly from the prediction with Boussinesq approximation.

\begin{figure}[htb]
    \includegraphics[width=\columnwidth]{acc_holes}
    \caption{
      Average acceleration of depletions for compressible and incompressible flows are shown.
      The continuous line shows the acceleration in Eq.~\eqref{eq:acceleration}
      with $\mathcal Q=0.32$
      while the dashed line is a linear reference line, which corresponds to the Boussinesq approximation.
    }
    \label{fig:acc_depletions}
\end{figure}
In Fig.~\ref{fig:acc_depletions} we show the simulated acceleration of depletions in the
compressible and the incompressible systems. We compare the simulation results
to the theoretical predictions Eq.~\eqref{eq:acceleration} of our model with and without inertia.
Deviations from our theoretical prediction Eq.~\eqref{eq:acceleration} are visible for amplitudes smaller than $\triangle n/n_0 \simeq -0.5$ (left of unity in the plot). The relative deviations are small at around $20$ percent.
As in Fig.~\ref{fig:com_depletions} the acceleration reaches a constant values
for plasma depletions of more than $90$ percent.
Comparing Fig.~\ref{fig:acc_depletions} to Fig.~\ref{fig:acc_blobs} the asymmetry between blobs and depletions becomes
apparent. While the acceleration of blobs is reduced for large
amplitudes compared to a linear dependence the acceleration
of depletions is increased. In the language of our simple buoyancy
model the inertia of depletions is reduced but increased for blobs.

In conclusion
  we discuss the dynamics of seeded blobs and depletions in a
  compressible and an incompressible system.
  With only two fit parameters our theoretical results reproduce the
  numerical COM velocities and accelerations over five orders of magnitude.
  We derive the amplitude dependence of the acceleration of blobs and depletions from
  the conservation laws of our systems in Eq.~\eqref{eq:acceleration}.
  From the same inequality a linear regime is derived in the compressible system for
  ratios of amplitudes to sizes smaller than a critical value.
   In this regime
  the blob and depletion velocity depends linearly on the initial amplitude and
  is independent of size. The regime is absent from the system with incompressible flows.
  Our theoretical results are verified by numerical simulations for all
  amplitudes that are relevant in magnetic fusion devices.
  Finally, we suggest a new empirical blob model that captures the detailed dynamics of more complicated models.
  The Boussinesq approximation is clarified as the absence of inertia and a thus altered acceleration of blobs and depletions.
  The maximum blob velocity is not altered by the Boussinesq approximation.

The authors were supported with financial subvention from the Research Council of Norway under grant
240510/F20. M.W. and M.H. were supported by the Austrian Science Fund (FWF) Y398.  The computational
results presented have been achieved in part using the Vienna Scientific Cluster (VSC). Part of this work was performed on the Abel Cluster, owned by the University of Oslo and the Norwegian metacenter
for High Performance Computing (NOTUR), and operated by the Department for Research Computing at USIT,
the University of Oslo IT-department.
This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.

Answer: The maximum velocity scales with the square root of the amplitude.


Input: What algorithm is engaged in the PLMS-PPIC method?

Context:
\section{Introduction}\label{S1}

The multiple access interferences (MAI) is the root of user
limitation in CDMA systems \cite{R1,R3}. The parallel least mean
square-partial parallel interference cancelation (PLMS-PPIC) method
is a multiuser detector for code division multiple access (CDMA)
receivers which reduces the effect of MAI in bit detection. In this
method and similar to its former versions like LMS-PPIC \cite{R5}
(see also \cite{RR5}), a weighted value of the MAI of other users is
subtracted before making the decision for a specific user in
different stages \cite{cohpaper}. In both of these methods, the
normalized least mean square (NLMS) algorithm is engaged
\cite{Haykin96}. The $m^{\rm th}$ element of the weight vector in
each stage is the true transmitted binary value of the $m^{\rm th}$
user divided by its hard estimate value from the previous stage. The
magnitude of all weight elements in all stages are equal to unity.
Unlike the LMS-PPIC, the PLMS-PPIC method tries to keep this
property in each iteration by using a set of NLMS algorithms with
different step-sizes instead of one NLMS algorithm used in LMS-PPIC.
In each iteration, the parameter estimate of the NLMS algorithm is
chosen whose element magnitudes of cancelation weight estimate have
the best match with unity. In PLMS-PPIC implementation it is assumed
that the receiver knows the phases of all user channels. However in
practice, these phases are not known and should be estimated. In
this paper we improve the PLMS-PPIC procedure \cite{cohpaper} in
such a way that when there is only a partial information of the
channel phases, this modified version simultaneously estimates the
phases and the cancelation weights. The partial information is the
quarter of each channel phase in $(0,2\pi)$.

The rest of the paper is organized as follows: In section \ref{S4}
the modified version of PLMS-PPIC with capability of channel phase
estimation is introduced. In section \ref{S5} some simulation
examples illustrate the results of the proposed method. Finally the
paper is concluded in section \ref{S6}.

\section{Multistage Parallel Interference Cancelation: Modified PLMS-PPIC Method}\label{S4}

We assume $M$ users synchronously send their symbols
$\alpha_1,\alpha_2,\cdots,\alpha_M$ via a base-band CDMA
transmission system where $\alpha_m\in\{-1,1\}$. The $m^{th}$ user
has its own code $p_m(.)$ of length $N$, where $p_m(n)\in \{-1,1\}$,
for all $n$. It means that for each symbol $N$ bits are transmitted
by each user and the processing gain is equal to $N$. At the
receiver we assume that perfect power control scheme is applied.
Without loss of generality, we also assume that the power gains of
all channels are equal to unity and users' channels do not change
during each symbol transmission (it can change from one symbol
transmission to the next one) and the channel phase $\phi_m$ of
$m^{th}$ user is unknown for all $m=1,2,\cdots,M$ (see
\cite{cohpaper} for coherent transmission). According to the above
assumptions the received signal is
\begin{equation}
\label{e1} r(n)=\sum\limits_{m=1}^{M}\alpha_m
e^{j\phi_m}p_m(n)+v(n),~~~~n=1,2,\cdots,N,
\end{equation}
where $v(n)$ is the additive white Gaussian noise with zero mean and
variance $\sigma^2$. Multistage parallel interference cancelation
method uses $\alpha^{s-1}_1,\alpha^{s-1}_2,\cdots,\alpha^{s-1}_M$,
the bit estimates outputs of the previous stage, $s-1$, to estimate
the related MAI of each user. It then subtracts it from the received
signal $r(n)$ and makes a new decision on each user variable
individually to make a new variable set
$\alpha^{s}_1,\alpha^{s}_2,\cdots,\alpha^{s}_M$ for the current
stage $s$. Usually the variable set of the first stage (stage $0$)
is the output of a conventional detector. The output of the last
stage is considered as the final estimate of transmitted bits. In
the following we explain the structure of a modified version of the
PLMS-PIC method \cite{cohpaper} with simultaneous capability of
estimating the cancelation weights and the channel phases.

Assume $\alpha_m^{(s-1)}\in\{-1,1\}$ is a given estimate of
$\alpha_m$ from stage $s-1$. Define
\begin{equation}
\label{e6} w^s_{m}=\frac{\alpha_m}{\alpha_m^{(s-1)}}e^{j\phi_m}.
\end{equation}
From (\ref{e1}) and (\ref{e6}) we have
\begin{equation}
\label{e7} r(n)=\sum\limits_{m=1}^{M}w^s_m\alpha^{(s-1)}_m
p_m(n)+v(n).
\end{equation}
Define
\begin{subequations}
\begin{eqnarray}
\label{e8} W^s&=&[w^s_{1},w^s_{2},\cdots,w^s_{M}]^T,\\
\label{e9}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!X^{s}(n)\!\!\!&=&\!\!\![\alpha^{(s-1)}_1p_1(n),\alpha^{(s-1)}_2p_2(n),\cdots,\alpha^{(s-1)}_Mp_M(n)]^T.
\end{eqnarray}
\end{subequations}
where $T$ stands for transposition. From equations (\ref{e7}),
(\ref{e8}) and (\ref{e9}), we have
\begin{equation}
\label{e10} r(n)=W^{s^T}X^{s}(n)+v(n).
\end{equation}
Given the observations $\{r(n),X^{s}(n)\}^{N}_{n=1}$, in modified
PLMS-PPIC, like the PLMS-PPIC \cite{cohpaper}, a set of NLMS
adaptive algorithm are used to compute
\begin{equation}
\label{te1} W^{s}(N)=[w^{s}_1(N),w^{s}_2(N),\cdots,w^{s}_M(N)]^T,
\end{equation}
which is an estimate of $W^s$ after iteration $N$. To do so, from
(\ref{e6}), we have
\begin{equation}
\label{e13} |w^s_{m}|=1 ~~~m=1,2,\cdots,M,
\end{equation}
which is equivalent to
\begin{equation}
\label{e14} \sum\limits_{m=1}^{M}||w^s_{m}|-1|=0.
\end{equation}
We divide $\Psi=\left(0,1-\sqrt{\frac{M-1}{M}}\right]$, a sharp
range for $\mu$ (the step-size of the NLMS algorithm) given in
\cite{sg2005}, into $L$ subintervals and consider $L$ individual
step-sizes $\Theta=\{\mu_1,\mu_2,\cdots,\mu_L\}$, where
$\mu_1=\frac{1-\sqrt{\frac{M-1}{M}}}{L}, \mu_2=2\mu_1,\cdots$, and
$\mu_L=L\mu_1$. In each stage, $L$ individual NLMS algorithms are
executed ($\mu_l$ is the step-size of the $l^{th}$ algorithm). In
stage $s$ and at iteration $n$, if
$W^{s}_k(n)=[w^s_{1,k},\cdots,w^s_{M,k}]^T$, the parameter estimate
of the $k^{\rm th}$ algorithm, minimizes our criteria, then it is
considered as the parameter estimate at time iteration $n$. In other
words if the next equation holds
\begin{equation}
\label{e17} W^s_k(n)=\arg\min\limits_{W^s_l(n)\in I_{W^s}
}\left\{\sum\limits_{m=1}^{M}||w^s_{m,l}(n)|-1|\right\},
\end{equation}
where $W^{s}_l(n)=W^{s}(n-1)+\mu_l \frac{X^s(n)}{\|X^s(n)\|^2}e(n),
~~~ l=1,2,\cdots,k,\cdots,L-1,L$ and
$I_{W^s}=\{W^s_1(n),\cdots,W^s_L(n)\}$, then we have
$W^s(n)=W^s_k(n)$, and therefore all other algorithms replace their
weight estimate by $W^{s}_k(n)$. At time instant $n=N$, this
procedure gives $W^s(N)$, the final estimate of $W^s$, as the true
parameter of stage $s$.

Now consider $R=(0,2\pi)$ and divide it into four equal parts
$R_1=(0,\frac{\pi}{2})$, $R_2=(\frac{\pi}{2},\pi)$,
$R_3=(\pi,\frac{3\pi}{2})$ and $R_4=(\frac{3\pi}{2},2\pi)$. The
partial information of channel phases (given by the receiver) is in
a way that it shows each $\phi_m$ ($m=1,2,\cdots,M$) belongs to
which one of the four quarters $R_i,~i=1,2,3,4$. Assume
$W^{s}(N)=[w^{s}_1(N),w^{s}_2(N),\cdots,w^{s}_M(N)]^T$ is the weight
estimate of the modified algorithm PLMS-PPIC at time instant $N$ of
the stage $s$. From equation (\ref{e6}) we have
\begin{equation}
\label{tt3}
\phi_m=\angle({\frac{\alpha^{(s-1)}_m}{\alpha_m}w^s_m}).
\end{equation}
We estimate $\phi_m$ by $\hat{\phi}^s_m$, where
\begin{equation}
\label{ee3}
\hat{\phi}^s_m=\angle{(\frac{\alpha^{(s-1)}_m}{\alpha_m}w^s_m(N))}.
\end{equation}
Because $\frac{\alpha^{(s-1)}_m}{\alpha_m}=1$ or $-1$, we have
\begin{eqnarray}
\hat{\phi}^s_m=\left\{\begin{array}{ll} \angle{w^s_m(N)} &
\mbox{if}~
\frac{\alpha^{(s-1)}_m}{\alpha_m}=1\\
\pm\pi+\angle{w^s_m(N)} & \mbox{if}~
\frac{\alpha^{(s-1)}_m}{\alpha_m}=-1\end{array}\right.
\end{eqnarray}
Hence $\hat{\phi}^s_m\in P^s=\{\angle{w^s_m(N)},
\angle{w^s_m(N)+\pi, \angle{w^s_m(N)}-\pi}\}$. If $w^s_m(N)$
sufficiently converges to its true value $w^s_m$, the same region
for $\hat{\phi}^s_m$ and $\phi_m$ is expected. In this case only one
of the three members of $P^s$ has the same region as $\phi_m$. For
example if $\phi_m \in (0,\frac{\pi}{2})$, then $\hat{\phi}^s_m \in
(0,\frac{\pi}{2})$ and therefore only $\angle{w^s_m(N)}$ or
$\angle{w^s_m(N)}+\pi$ or $\angle{w^s_m(N)}-\pi$ belongs to
$(0,\frac{\pi}{2})$. If, for example, $\angle{w^s_m(N)}+\pi$ is such
a member between all three members of $P^s$, it is the best
candidate for phase estimation. In other words,
\[\phi_m\approx\hat{\phi}^s_m=\angle{w^s_m(N)}+\pi.\]
We admit that when there is a member of $P^s$ in the quarter of
$\phi_m$, then $w^s_m(N)$ converges. What would happen when non of
the members of $P^s$ has the same quarter as $\phi_m$? This
situation will happen when the absolute difference between $\angle
w^s_m(N)$ and $\phi_m$ is greater than $\pi$. It means that
$w^s_m(N)$ has not converged yet. In this case where we can not
count on $w^s_m(N)$, the expected value is the optimum choice for
the channel phase estimation, e.g. if $\phi_m \in (0,\frac{\pi}{2})$
then $\frac{\pi}{4}$ is the estimation of the channel phase
$\phi_m$, or if $\phi_m \in (\frac{\pi}{2},\pi)$ then
$\frac{3\pi}{4}$ is the estimation of the channel phase $\phi_m$.
The results of the above discussion are summarized in the next
equation
\begin{eqnarray}
\nonumber \hat{\phi}^s_m = \left\{\begin{array}{llll} \angle
{w^s_m(N)} & \mbox{if}~
\angle{w^s_m(N)}, \phi_m\in R_i,~~i=1,2,3,4\\
\angle{w^s_m(N)}+\pi & \mbox{if}~ \angle{w^s_m(N)}+\pi, \phi_m\in
R_i,~~i=1,2,3,4\\
\angle{w^n_m(N)}-\pi & \mbox{if}~ \angle{w^s_m(N)}-\pi, \phi_m\in
R_i,~~i=1,2,3,4\\
\frac{(i-1)\pi+i\pi}{4} & \mbox{if}~ \phi_m\in
R_i,~~\angle{w^s_m(N)},\angle
{w^s_m(N)}\pm\pi\notin R_i,~~i=1,2,3,4.\\
\end{array}\right.
\end{eqnarray}
Having an estimation of the channel phases, the rest of the proposed
method is given by estimating $\alpha^{s}_m$ as follows:
\begin{equation}
\label{tt4}
\alpha^{s}_m=\mbox{sign}\left\{\mbox{real}\left\{\sum\limits_{n=1}^{N}
q^s_m(n)e^{-j\hat{\phi}^s_m}p_m(n)\right\}\right\},
\end{equation}
where
\begin{equation} \label{tt5}
q^{s}_{m}(n)=r(n)-\sum\limits_{m^{'}=1,m^{'}\ne
m}^{M}w^{s}_{m^{'}}(N)\alpha^{(s-1)}_{m^{'}} p_{m^{'}}(n).
\end{equation}
The inputs of the first stage $\{\alpha^{0}_m\}_{m=1}^M$ (needed for
computing $X^1(n)$) are given by
\begin{equation}
\label{qte5}
\alpha^{0}_m=\mbox{sign}\left\{\mbox{real}\left\{\sum\limits_{n=1}^{N}
r(n)e^{-j\hat{\phi}^0_m}p_m(n)\right\}\right\}.
\end{equation}
Assuming $\phi_m\in R_i$, then
\begin{equation}
\label{qqpp} \hat{\phi}^0_m =\frac{(i-1)\pi+i\pi}{4}.
\end{equation}
Table \ref{tab4} shows the structure of the modified PLMS-PPIC
method. It is to be notified that
\begin{itemize}
\item Equation (\ref{qte5}) shows the conventional bit detection
method when the receiver only knows the quarter of channel phase in
$(0,2\pi)$. \item With $L=1$ (i.e. only one NLMS algorithm), the
modified PLMS-PPIC can be thought as a modified version of the
LMS-PPIC method.
\end{itemize}

In the following section some examples are given to illustrate the
effectiveness of the proposed method.

\section{Simulations}\label{S5}

In this section we have considered some simulation examples.
Examples \ref{ex2}-\ref{ex4} compare the conventional, the modified
LMS-PPIC and the modified PLMS-PPIC methods in three cases: balanced
channels, unbalanced channels and time varying channels. In all
examples, the receivers have only the quarter of each channel phase.
Example \ref{ex2} is given to compare the modified LMS-PPIC and the
PLMS-PPIC in the case of balanced channels.

\begin{example}{\it Balanced channels}:
\label{ex2}
\begin{table}
\caption{Channel phase estimate of the first user (example
\ref{ex2})} \label{tabex5} \centerline{{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{6}{*}{\rotatebox{90}{$\phi_m=\frac{3\pi}{8},M=15~~$}} & N(Iteration) & Stage Number& NLMS & PNLMS  \\
&&&&\\
\cline{2-5} & \multirow{2}{*}{64}& s = 2 &  $\hat{\phi}^s_m=\frac{3.24\pi}{8}$ & $\hat{\phi}^s_m=\frac{3.18\pi}{8}$ \\
\cline{3-5} & & s = 3 & $\hat{\phi}^s_m=\frac{3.24\pi}{8}$ & $\hat{\phi}^s_m=\frac{3.18\pi}{8}$ \\
\cline{2-5} & \multirow{2}{*}{256}& s = 2 &  $\hat{\phi}^s_m=\frac{2.85\pi}{8}$ & $\hat{\phi}^s_m=\frac{2.88\pi}{8}$ \\
\cline{3-5} & & s = 3 & $\hat{\phi}^s_m=\frac{2.85\pi}{8}$ & $\hat{\phi}^s_m=\frac{2.88\pi}{8}$ \\
\cline{2-5} \hline
\end{tabular} }}
\end{table}
Consider the system model (\ref{e7}) in which $M$ users
synchronously send their bits to the receiver through their
channels. It is assumed that each user's information consists of
codes of length $N$. It is also assumd that the signal to noise
ratio (SNR) is 0dB. In this example there is no power-unbalanced or
channel loss is assumed. The step-size of the NLMS algorithm in
modified LMS-PPIC method is $\mu=0.1(1-\sqrt{\frac{M-1}{M}})$ and
the set of step-sizes of the parallel NLMS algorithms in modified
PLMS-PPIC method are
$\Theta=\{0.01,0.05,0.1,0.2,\cdots,1\}(1-\sqrt{\frac{M-1}{M}})$,
i.e. $\mu_1=0.01(1-\sqrt{\frac{M-1}{M}}),\cdots,
\mu_4=0.2(1-\sqrt{\frac{M-1}{M}}),\cdots,
\mu_{12}=(1-\sqrt{\frac{M-1}{M}})$. Figure~\ref{Figexp1NonCoh}
illustrates the bit error rate (BER) for the case of two stages and
for $N=64$ and $N=256$. Simulations also show that there is no
remarkable difference between results in two stage and three stage
scenarios. Table~\ref{tabex5} compares the average channel phase
estimate of the first user in each stage and over $10$ runs of
modified LMS-PPIC and PLMS-PPIC, when the the number of users is
$M=15$.
\end{example}

Although LMS-PPIC and PLMS-PPIC, as well as their modified versions,
are structured based on the assumption of no near-far problem
(examples \ref{ex3} and \ref{ex4}), these methods and especially the
second one have remarkable performance in the cases of unbalanced
and/or time varying channels.

\begin{example}{\it Unbalanced channels}:
\label{ex3}
\begin{table}
\caption{Channel phase estimate of the first user (example
\ref{ex3})} \label{tabex6} \centerline{{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{6}{*}{\rotatebox{90}{$\phi_m=\frac{3\pi}{8},M=15~~$}} & N(Iteration) & Stage Number& NLMS & PNLMS  \\
&&&&\\
\cline{2-5} & \multirow{2}{*}{64}& s=2 &  $\hat{\phi}^s_m=\frac{2.45\pi}{8}$ & $\hat{\phi}^s_m=\frac{2.36\pi}{8}$ \\
\cline{3-5} & & s=3 & $\hat{\phi}^s_m=\frac{2.71\pi}{8}$ & $\hat{\phi}^s_m=\frac{2.80\pi}{8}$ \\
\cline{2-5} & \multirow{2}{*}{256}& s=2 &  $\hat{\phi}^s_m=\frac{3.09\pi}{8}$ & $\hat{\phi}^s_m=\frac{2.86\pi}{8}$ \\
\cline{3-5} & & s=3 & $\hat{\phi}^s_m=\frac{2.93\pi}{8}$ & $\hat{\phi}^s_m=\frac{3.01\pi}{8}$ \\
\cline{2-5} \hline
\end{tabular} }}
\end{table}
Consider example \ref{ex2} with power unbalanced and/or channel loss
in transmission system, i.e. the true model at stage $s$ is
\begin{equation}
\label{ve7} r(n)=\sum\limits_{m=1}^{M}\beta_m
w^s_m\alpha^{(s-1)}_m c_m(n)+v(n),
\end{equation}
where $0<\beta_m\leq 1$ for all $1\leq m \leq M$. Both the LMS-PPIC
and the PLMS-PPIC methods assume the model (\ref{e7}), and their
estimations are based on observations $\{r(n),X^s(n)\}$, instead of
$\{r(n),\mathbf{G}X^s(n)\}$, where the channel gain matrix is
$\mathbf{G}=\mbox{diag}(\beta_1,\beta_2,\cdots,\beta_m)$. In this
case we repeat example \ref{ex2}. We randomly get each element of
$G$ from $[0,0.3]$. Figure~\ref{Figexp2NonCoh} illustrates the BER
versus the number of users. Table~\ref{tabex6} compares the channel
phase estimate of the first user in each stage and over $10$ runs of
modified LMS-PPIC and modified PLMS-PPIC for $M=15$.
\end{example}

\begin{example}
\label{ex4} {\it Time varying channels}: Consider example \ref{ex2}
with time varying Rayleigh fading channels. In this case we assume
the maximum Doppler shift of $40$HZ, the three-tap
frequency-selective channel with delay vector of $\{2\times
10^{-6},2.5\times 10^{-6},3\times 10^{-6}\}$sec and gain vector of
$\{-5,-3,-10\}$dB. Figure~\ref{Figexp3NonCoh} shows the average BER
over all users versus $M$ and using two stages.
\end{example}

\section{Conclusion}\label{S6}

In this paper, parallel interference cancelation using adaptive
multistage structure and employing a set of NLMS algorithms with
different step-sizes is proposed, when just the quarter of the
channel phase of each user is known. In fact, the algorithm has been
proposed for coherent transmission with full information on channel
phases in \cite{cohpaper}. This paper is a modification on the
previously proposed algorithm. Simulation results show that the new
method has a remarkable performance for different scenarios
including Rayleigh fading channels even if the channel is
unbalanced.

Answer: The normalized least mean square (NLMS) algorithm.


Input: What is the significance of the interlayer Berry connection polarizability?

Context:
Paper Info

Title: Crossed Nonlinear Dynamical Hall Effect in Twisted Bilayers
Publish Date: 17 Mar 2023
Author List:

Figure

FIG. 1.(a) Schematics of experimental setup.(b, c) Valence band structure and intrinsic Hall conductivity with respect to in-plane input for tMoTe2 at twist angles (b) θ = 1.2 • and (c) θ = 2 • in +K valley.Color coding in (b) and (c) denotes the layer composition σ z n (k).
FIG. 2. (a) The interlayer BCP G, and (b) its vorticity [∂ k × G]z on the first valence band from +K valley of 1.2 • tMoTe2.Background color and arrows in (a) denote the magnitude and vector flow, respectively.Grey curves in (b) show energy contours at 1/2 and 3/4 of the band width.The black dashed arrow denotes direction of increasing hole doping level.Black dashed hexagons in (a, b) denote the boundary of moiré Brillouin zone (mBZ).
FIG. 3. (a-c) Three high-symmetry stacking registries for tBG with a commensurate twist angle θ = 21.8 • .Lattice geometries with rotation center on an overlapping atomic site (a, b) and hexagonal center (c).(d) Schematic of the moiré pattern when the twist angle slightly deviates from 21.8 • , here θ = 21 • .Red squares marked by A, B and C are the local regions that resemble commensurate 21.8 • patterns in (a), (b) and (c), respectively.(e, f) Low-energy band structures and intrinsic Hall conductivity of the two geometries [(a) and (b) are equivalent].The shaded areas highlight energy windows ∼ ω around band degeneracies where interband transitions, not considered here, may quantitatively affect the conductivity measured.
FIG. S4.Band structure and layer composition σ z n in +K valley of tBG (left panel) and the intrinsic Hall conductivity (right panel) at three different twist angle θ.The shaded areas highlight energy windows ∼ ω around band degeneracies in which the conductivity results should not be considered.Here σH should be multiplied by a factor of 2 accounting for spin degeneracy.

abstract

We propose an unconventional nonlinear dynamical Hall effect characteristic of twisted bilayers. The joint action of in-plane and out-of-plane ac electric fields generates Hall currents j ∼ Ė⊥ × E in both sum and difference frequencies, and when the two orthogonal fields have common frequency their phase difference controls the on/off, direction and magnitude of the rectified dc Hall current.
This novel intrinsic Hall response has a band geometric origin in the momentum space curl of interlayer Berry connection polarizability, arising from layer hybridization of electrons by the twisted interlayer coupling. The effect allows a unique rectification functionality and a transport probe of chiral symmetry in bilayer systems.
We show sizable effects in twisted homobilayer transition metal dichalcogenides and twisted bilayer graphene over broad range of twist angles. Nonlinear Hall-type response to an in-plane electric field in a two dimensional (2D) system with time reversal symmetry has attracted marked interests . Intensive studies have been devoted to uncovering new types of nonlinear Hall transport induced by quantum geometry and their applications such as terahertz rectification and magnetic information readout .
Restricted by symmetry , the known mechanisms of nonlinear Hall response in quasi-2D nonmagnetic materials are all of extrinsic nature, sensitive to fine details of disorders , which have limited their utilization for practical applications. Moreover, having a single driving field only, the effect has not unleashed the full potential of nonlinearity for enabling controlled gate in logic operation, where separable inputs (i.e., in orthogonal directions) are desirable.
The latter, in the context of Hall effect, calls for control by both out-of-plane and in-plane electric fields. A strategy to introduce quantum geometric response to out-of-plane field in quasi-2D geometry is made possible in van der Waals (vdW) layered structures with twisted stacking . Taking homobilayer as an example, electrons have an active layer degree of freedom that is associated with an out-of-plane electric dipole , whereas interlayer quantum tunneling rotates this pseudospin about in-plane axes that are of topologically nontrivial textures in the twisted landscapes .
Such layer pseudospin structures can underlie novel quantum geometric properties when coupled with out-ofplane field. Recent studies have found layer circular photogalvanic effect and layer-contrasted time-reversaleven Hall effect , arising from band geometric quantities. In this work we unveil a new type of nonlinear Hall effect in time-reversal symmetric twisted bilayers, where an intrinsic Hall current emerges under the combined action of an in-plane electric field E and an out-of-plane ac field E ⊥ (t): j ∼ Ė⊥ × E [see Fig. ].
Having the two driving fields (inputs) and the current response (output) all orthogonal to each other, the effect is dubbed as the crossed nonlinear dynamical Hall effect. This is also the first nonlinear Hall contribution of an intrinsic nature in nonmagnetic materials without external magnetic field, determined solely by the band structures, not relying on extrinsic factors such as disorders and relaxation times.
The effect arises from the interlayer hybridization of electronic states under the chiral crystal symmetry characteristic of twisted bilayers, and has a novel band geometric origin in the momentum space curl of interlayer Berry connection polarizability (BCP). Having two driving fields of the same frequency, a dc Hall current develops, whose on/off, direction and magnitude can all be controlled by the phase difference of the two fields, which does not affect the magnitude of the double-frequency component.
Such a characteristic tunability renders this effect a unique approach to rectification and transport probe of chiral bilayers. As examples, we show sizable effects in small angle twisted transition metal dichalcogenides (tTMDs) and twisted bilayer graphene (tBG), as well as tBG of large angles where Umklapp interlayer tunneling dominates.
Geometric origin of the effect. A bilayer system couples to in-plane and out-of-plane driving electric fields in completely different ways. The in-plane field couples to the 2D crystal momentum, leading to Berry-phase effects in the 2D momentum space . In comparison, the outof-plane field is coupled to the interlayer dipole moment p in the form of −E ⊥ p, where p = ed 0 σz with σz as the Pauli matrix in the layer index subspace and d 0 the interlayer distance.
When the system has a more than twofold rotational axis in the z direction, as in tBG and tTMDs, any in-plane current driven by the out-of-plane field alone is forbidden. It also prohibits the off-diagonal components of the symmetric part of the conductivity tensor σ ab = ∂j a /∂E ||,b with respect to the in-plane input and output.
Since the antisymmetric part of σ ab is not allowed by the Onsager reciprocity in nonmagnetic systems, all the off-diagonal components of σ ab is forbidden, irrespective of the order of out-of-plane field. On the other hand, as we will show, an in-plane Hall conductivity σ xy = −σ yx can still be driven by the product of an in-plane field and the time variation rate of an outof-plane ac field, which is a characteristic effect of chiral bilayers.
To account for the effect, we make use of the semiclassical theory . The velocity of an electron in a bilayer system is given by where k is the 2D crystal momentum. Here and hereafter we suppress the band index for simplicity, unless otherwise noted. The three contributions in this equation come from the band velocity, the anomalous velocities induced by the k -space Berry curvature Ω k and by the hybrid Berry curvature Ω kE ⊥ in the (k, E ⊥ ) space.
For the velocity at the order of interest, the k-space Berry curvature is corrected to the first order of the variation rate of out-of-plane field Ė⊥ as Here A = u k |i∂ k |u k is the unperturbed k-space Berry connection, with |u k being the cell-periodic part of the Bloch wave, whereas is its gauge invariant correction , which can be identified physically as an in-plane positional shift of an electron induced by the time evolution of the out-of-plane field.
For a band with index n, we have whose numerator involves the interband matrix elements of the interlayer dipole and velocity operators, and ε n is the unperturbed band energy. Meanwhile, up to the first order of in-plane field, the hybrid Berry curvature reads Here A E || is the k-space Berry connection induced by E || field , which represents an intralayer positional shift and whose detailed expression is not needed for our purpose.
and is its first order correction induced by the in-plane field. In addition, ε = ε + δε, where δε = eE • G Ė⊥ is the field-induced electron energy . Given that A E || is the E ⊥ -space counterpart of intralayer shift A E || , and that E ⊥ is conjugate to the interlayer dipole moment, we can pictorially interpret A E || as the interlayer shift induced by in-plane field.
It indeed has the desired property of flipping sign under the horizontal mirror-plane reflection, hence is analogous to the so-called interlayer coordinate shift introduced in the study of layer circular photogalvanic effect , which is nothing but the E ⊥ -space counterpart of the shift vector well known in the nonlinear optical phenomenon of shift current.
Therefore, the E ⊥ -space BCP eG/ can be understood as the interlayer BCP. This picture is further augmented by the connotation that the interlayer BCP is featured exclusively by interlayer-hybridized electronic states: According to Eq. ( ), if the state |u n is fully polarized in a specific layer around some momentum k, then G (k) is suppressed.
With the velocity of individual electrons, the charge current density contributed by the electron system can be obtained from where [dk] is shorthand for n d 2 k/(2π) 2 , and the distribution function is taken to be the Fermi function f 0 as we focus on the intrinsic response. The band geometric contributions to ṙ lead to a Hall current
where is intrinsic to the band structure. This band geometric quantity measures the k-space curl of the interlayer BCP over the occupied states, and hence is also a characteristic of layer-hybridized electronic states. Via an integration by parts, it becomes clear that χ int is a Fermi surface property.
Since χ int is a time-reversal even pseudoscalar, it is invariant under rotation, but flips sign under space inversion, mirror reflection and rotoreflection symmetries. As such, χ int is allowed if and only if the system possesses a chiral crystal structure, which is the very case of twisted bilayers .
Moreover, since twisted structures with opposite twist angles are mirror images of each other, whereas the mirror reflection flips the sign of χ int , the direction of Hall current can be reversed by reversing twist direction. Hall rectification and frequency doubling. This effect can be utilized for the rectification and frequency doubling of an in-plane ac input E = E 0 cos ωt, provided that the out-of-plane field has the same frequency, namely E ⊥ = E 0 ⊥ cos (ωt + ϕ).
The phase difference ϕ between the two fields plays an important role in determining the Hall current, which takes the form of j = j 0 sin ϕ + j 2ω sin(2ωt + ϕ). ( Here ω is required to be below the threshold for direct interband transition in order to validate the semiclassical treatment, and σ H has the dimension of conductance and quantifies the Hall response with respect to the in-plane input.
In experiment, the Hall output by the crossed nonlinear dynamic Hall effect can be distinguished readily from the conventional nonlinear Hall effect driven by in-plane field alone, as they are odd and even, respectively, in the inplane field. One notes that while the double-frequency component appears for any ϕ, the rectified output is allowed only if the two crossed driving fields are not in-phase or antiphase.
Its on/off, chirality (right or left), and magnitude are all controlled by the phase difference of the two fields. Such a unique tunability provides not only a prominent experimental hallmark of this effect, but also a controllable route to Hall rectification. In addition, reversing the direction of the out-of-plane field switches that of the Hall current, which also serves as a control knob.
Application to tTMDs. We now study the effect quantitatively in tTMDs, using tMoTe 2 as an example (see details of the continuum model in ). For illustrative purposes, we take ω/2π = 0.1 THz and E 0 ⊥ d 0 = 10 mV in what follows. Figures ) and (c) present the electronic band structures along with the layer composition σ z n (k) at twist angles θ = 1.2 • and θ = 2 • .
In both cases, the energy spectra exhibit isolated narrow bands with strong layer hybridization. At θ = 1.2 • , the conductivity shows two peaks ∼ 0.1e 2 /h at low energies associated with the first two valence bands. The third band does not host any sizable conductivity signal. At higher hole-doping levels, a remarkable conductivity peak ∼ e 2 /h appears near the gap separating the fourth and fifth bands.
At θ = 2 • , the conductivity shows smaller values, but the overall trends are similar: A peak ∼ O(0.01)e 2 /h appears at low energies, while larger responses ∼ O(0.1)e 2 /h can be spotted as the Fermi level decreases. One can understand the behaviors of σ H from the interlayer BCP in Eq. ( ). It favors band near-degeneracy regions in k -space made up of strongly layer hybridized electronic states.
As such, the conductivity is most pro- nounced when the Fermi level is located around such regions, which directly accounts for the peaks of response in Fig. that [∂ k × G] z is negligible at lower energies, and it is dominated by positive values as the doping increases, thus the conductivity rises initially.
When the doping level is higher, regions with [∂ k × G] z < 0 start to contribute, thus the conductivity decreases after reaching a maximum. Application to tBG. The second example is tBG. We focus on commensurate twist angles in the large angle limit in the main text , which possess moiré-lattice assisted strong interlayer tunneling via Umklapp processes .
This case is appealing because the Umklapp interlayer tunneling is a manifestation of discrete translational symmetry of moiré superlattice, which is irrelevant at small twist angles and not captured by the continuum model but plays important roles in physical contexts such as higher order topological insulator and moiré excitons .
The Umklapp tunneling is strongest for the commensurate twist angles of θ = 21.8 • and θ = 38.2 • , whose corresponding periodic moiré superlattices have the smallest lattice constant ( √ 7 of the monolayer counterpart). Such a small moiré scale implies that the exact crystalline symmetry, which depends sensitively on fine details of rotation center, has critical influence on lowenergy response properties.
To capture the Umklapp tunneling, we employ the tight-binding model . Figures ) and (c) show two distinct commensurate structures of tBG at θ = 21.8 • belonging to chiral point groups D 3 and D 6 , respectively. The atomic configurations in Figs. ) are equivalent, which are constructed by twisting AA-stacked bilayer graphene around an overlapping atom site, and that in Fig. ) is obtained by rotating around a hexagonal center.
Band structures of these two configurations are drastically different within a low-energy window of ∼ 10 meV around the κ point . Remarkably, despite large θ, we still get σ H ∼ O(0.001) e 2 /h (D 3 ) and ∼ O(0.1) e 2 /h (D 6 ), which are comparable to those at small angles (cf. Fig. in the Supplemental Material ).
Such sizable responses can be attributed to the strong interlayer coupling enabled by Umklapp processes . Apart from different intensities, the Hall conductivities in the two stacking configurations have distinct energy dependence: In Fig. , σ H shows a single peak centered at zero energy; In Fig. (f), it exhibits two antisymmetric peaks around zero.
The peaks are centered around band degeneracies, and their profiles can be understood from the distribution of [∂ k × G] z . Figure (d) illustrates the atomic structure of tBG with a twist angle slightly deviating from θ = 21.8 • , forming a supermoiré pattern. In short range, the local stacking geometries resemble the commensurate configurations at θ = 21.8 • , while the stacking registries at different locales differ by a translation.
Similar to the moiré landscapes in the small-angle limit, there also exist high-symmetry locales: Regions A and B enclose the D 3 structure, and region C contains the D 6 configuration. Position-dependent Hall response is therefore expected in such a supermoiré. As the intrinsic Hall signal from the D 6 configuration dominates [see Figs.
3(e) vs (f)], the net response mimics that in Fig. . Discussion. We have uncovered the crossed nonlinear dynamical intrinsic Hall effect characteristic of layer hybridized electronic states in twisted bilayers, and elucidated its geometric origin in the k -space curl of interlayer BCP. It offers a new tool for rectification and frequency doubling in chiral vdW bilayers, and is sizable in tTMD and tBG.
Here our focus is on the intrinsic effect, which can be evaluated quantitatively for each material and provides a benchmark for experiments. There may also be extrinsic contributions, similar to the side jump and skew scattering ones in anomalous Hall effect. They typically have distinct scaling behavior with the relaxation time τ from the intrinsic effect, hence can be distinguished from the latter in experiments .
Moreover, they are suppressed in the clean limit ωτ 1 [(ωτ ) 2 1, more precisely] . In high-quality tBG samples, τ ∼ ps at room temperature . Much longer τ can be obtained at lower temperatures. In fact, a recent theory explaining well the resistivity of tBG predicted τ ∼ 10 −8 s at 10 K . As such, high-quality tBG under low temperatures and sub-terahertz input (ω/2π = 0.1 THz) is located in the clean limit, rendering an ideal platform for isolating the intrinsic effect.
This work paves a new route to driving in-plane response by out-of-plane dynamical control of layered vdW structures . The study can be generalized to other observables such as spin current and spin polarization, and the in-plane driving can be statistical forces, like temperature gradient. Such orthogonal controls rely critically on the nonconservation of layer pseudospin degree of freedom endowed by interlayer coupling, and constitute an emerging research field at the crossing of 2D vdW materials, layertronics, twistronics and nonlinear electronics.
This work is supported by the Research Grant Council of Hong Kong (AoE/P-701/20, HKU SRFS2122-7S05), and the Croucher Foundation. W.Y. also acknowledges support by Tencent Foundation. Cong Chen, 1, 2, * Dawei Zhai, 1, 2, * Cong Xiao, 1, 2, † and Wang Yao 1, 2, ‡ 1 Department of Physics, The University of Hong Kong, Hong Kong, China 2 HKU-UCAS Joint Institute of Theoretical and Computational Physics at Hong Kong, China Extra figures for tBG at small twist angles Figure (a) shows the band structure of tBG with θ = 1.47 • obtained from the continuum model .
The central bands are well separated from higher ones, and show Dirac points at κ/κ points protected by valley U (1) symmetry and a composite operation of twofold rotation and time reversal C 2z T . Degeneracies at higher energies can also be identified, for example, around ±75 meV at the γ point. As the two Dirac cones from the two layers intersect around the same area, such degeneracies are usually accompanied by strong layer hybridization [see the color in the left panel of Fig. ].
Additionally, it is well-known that the two layers are strongly coupled when θ is around the magic angle (∼ 1.08 • ), rendering narrow bandwidths for the central bands. As discussed in the main text, coexistence of strong interlayer hybridization and small energy separations is expected to contribute sharp conductivity peaks near band degeneracies, as shown in Fig. .
In this case, the conductivity peak near the Dirac point can reach ∼ 0.1e 2 /h, while the responses around ±0.08 eV are smaller at ∼ 0.01e 2 /h. The above features are maintained when θ is enlarged, as illustrated in Figs. ) and (c) using θ = 2.65 • and θ = 6.01 • . Since interlayer coupling becomes weaker and the bands are more separated at low energies when θ increases, intensity of the conductivity drops significantly.
We stress that G is not defined at degenerate points, and interband transitions may occur when energy separation satisfies |ε n − ε m | ∼ ω, the effects of which are not included in the current formulations. Consequently, the results around band degeneracies within energy ∼ ω [shaded areas in Fig. ] should be excluded.

Answer: The momentum space curl of the interlayer Berry connection polarizability generates the crossed nonlinear dynamical Hall effect.