Why There's No Quick Fix For China's Ailing Property Market
China’s top housing official has stepped up rhetoric meant to revive the housing market. It comes after the Politburo removed “the housing is not for speculation” slogan from the readout of its meeting, which increased expectations for more support for the market. Unfortunately, there’s no panacea to end the crisis quickly.
Hang Seng futures pointed to a weaker opening Friday. Strong US data spurred a dollar rally and higher US Treasury yields, which may weigh on foreign inflows to China. A Nikkei report that the Bank of Japan may discuss changing the yield-curve control policy added to uncertainties.
On the China front, the news flow continues a pattern of traders going “long on the words, short on actions.” Top housing official on Thursday urged more support, including calling for homebuyers who had paid off previous mortgages to be considered as first-time purchasers, so that they could enjoy lower mortgage rates. (The so-called “recognizing houses but not loans” policy.）
None of the talking points are entirely new. In 2022, 57 cities have adopted the “recognizing houses” policy, according to Nomura, citing data from China Real Estate Information Corp. Altogether, nearly 300 cities issued almost 600 various easing measures last year, including lowering down payments and loosening purchasing restrictions.
If that hasn’t helped prop up the market already, one can be excused for having doubt that any incremental, piecemeal measures will do the trick.
In a report published in June, Nomura’s economists, including Lu Ting, listed a few reasons why investors should lower their expectations on the housing stimulus, even though more support is likely to come.
For starters, Beijing simply has no appetite for a policy bazooka when the priority is focused on security and sustainability. So forget about another round “shantytown renovation” programs. That scheme, which offered cash compensation for homes demolished in less-developed areas, helped turn around a housing downturn in 2015-2016, but it also helped fueled a real estate bubble in lower-tier cities.
Second, some easing measures will likely increase sales of existing homes, strengthening expectations of home price declines and delaying purchases.
It’s questionable that China will meaningfully ease restrictions in big cities such as Beijing and Shanghai. Even if it does, easing in big cities may crowd out the demand for homes in low-tier cities, which have been the driver of commodity demand and construction activity over the past decade.
Smaller cities are still suffering from the overhang of the shantytown renovations, which have pulled forward home demand. These cities are facing high leverage, falling home prices and population outflows. Coupled with a large amount of unfinished projects and the withdrawal of private developers, a sustainable property rebound there is questionable.
Finally, the capability and willingness of Chinese households to borrow and buy homes may have been significantly reduced, even in large cities, once expectations that housing prices can only go up have been shattered.
All told, an “L-shaped” recovery in housing is all one can hope for.
Soaring Miu Miu Sales Boost Prada Group
Miu Miu’s first-half revenues jumped 50 percent year on year as runway relevance translated into sales. The group’s flagship Prada brand grew 18 percent.
Prada’s sister brand Miu Miu reported a historic first half — with like-for-like retail sales jumping by 50 percent — as the label’s renewed runway buzz translated into a soaring revenues. Sales at its flagship Prada brand grew 18 percent, the Milanese fashion group said.
In a call with analysts, Prada’s revamped management team cited the cultural relevance and clarified positioning of Miu Miu as a key growth driver, as well as the brand’s exposure to Asian markets where sales are rebounding from coronavirus shutdowns last year).
In October 2021, the company unveiled a more legible, social media-friendly brand platform for Miu Miu, focusing on rebellious twists on classic collegiate dress, which has pushed interest in the label to new heights. Miu Miu has since continued to develop those themes, deploying ironically ladylike officewear, ballet flats, luxe tights and retro pedal pushers. A revamped handbag program has invoked the early 2000s It-bag aesthetic but with a timeless touch that helped keep items from feeling overly trendy.
Prada Group’s first-half revenues of €2.2 billion rose 20 percent year-on-year, slightly missing analyst estimates. Sales rose sharply among Asian customers, while Europeans “normalised, but remain very solid,” chief executive officer Andrea Guerra said.
Like many competitors, Prada Group is facing slowing demand in the key US market, where its retail sales fell 1 percent in the first half. Still, the group said its business with American clients continues to grow overall, with a low-single digit sales increase once sales to US tourists in Europe were taken into account.
Last week, declining sales at Cartier-owner Richemont sent its shares tumbling as much as 10 percent in one day. Earlier this week sector leader LVMH reported similar weakness in the US, citing lower demand from aspirational customers.
Prada Group’s strong results — nearly in line with LVMH’s 20 percent fashion second-quarter growth — suggest that the company has the necessary momentum to navigate an increasingly uncertain fashion market without the backing of a major conglomerate.
The company recently recast its executive ranks in a bid to bolster governance and ease the transition to its next generation of leaders: While controlling shareholder Miuccia Prada remains co-creative director alongside Raf Simons, she relinquished her co-CEO title upon the arrival of former Luxottica chief Guerra earlier this year. Her husband Patrizio Bertelli also relinquished his CEO title, transitioning to an executive chairman role. Additionally, the group named a new brand-level chief executive at Prada, Gianfranco D’Attis, formerly of Dior.
Kering Buys 30% Stake in Valentino, Signaling New Strategy
The French luxury group announced a broader partnership with Mayhoola as it charts a course for growth.
PARIS — Kering said on Thursday it has bought a 30 percent stake in Valentino for 1.7 billion euros in cash as part of a broader strategic partnership with Qatari investment fund Mayhoola, as it seeks to chart a course for growth during a transitional year for its star brand Gucci.
The acquisition, which came a week after a major management reshuffle, overshadowed a weak performance in the second quarter that saw Gucci miss market expectations as it adapts to a new corporate and creative leadership following the exit of longtime chief executive officer Marco Bizzarri and creative director Alessandro Michele.
Kering has an option to buy 100 percent of Valentino’s capital by 2028, while Mayhoola could become a shareholder in Kering, the brands said in a joint statement released after the Paris market close. The initial transaction for the French luxury group to buy its 30 percent Valentino stake is expected to close by the end of 2023, subject to clearance by the relevant competition authorities.
François-Henri Pinault, chairman and CEO of Kering, said Valentino would benefit from synergies, while Kering and Mayhoola will jointly explore further opportunities aligned with their respective strategies.
“It’s too early, of course, to say what would be those opportunities, but for me, it has to broaden our scope of looking at the luxury markets in the future,” he said. “Over the past 10 years, we’ve made great progress in becoming an integrated luxury group. The transformation, however, is not yet complete.”
Pinault said Valentino fills a gap in Kering’s portfolio as the group targets wealthier customers with high-end products such as exotic leather goods and high jewelry, after tripling in size over the last decade. He said Valentino CEO Jacopo Venturini, who joined the label in 2020 from Gucci, would remain in place.
“Valentino is a house that I’ve always admired. It’s an amazing Italian name rooted in haute couture and known all around the world, so we’re very proud to be able to support the brand elevation strategy successfully implemented in the past few years,” Pinault said.
The executive made no specific mention of Valentino creative director Pierpaolo Piccioli.
Rachid Mohamed Rachid, CEO of Mayhoola and chairman of Valentino, said it looked forward to joining forces with Kering, whose brands also include Saint Laurent, Bottega Veneta and Balenciaga.
“Under our stewardship, Valentino has strengthened its foundations as a highly desirable luxury brand and we will keep reinforcing the brand in the next chapter with Kering. We look forward to our partnership with Kering in Valentino and also in other potential opportunities to explore investments together,” he said in the statement.
Valentino has 211 directly operated stores in more than 25 countries and posted revenues of 1.4 billion euros and recurring earnings before interest, taxes, depreciation and amortization of 350 million euros in 2022, according to the statement.
“The potential of this brand is, in our opinion, quite significant for the next years to come,” Pinault said. “The discussion we had with the management of Valentino during the process of the acquisition reinforced our confidence in what we could bring to Valentino in the coming years to not only continue to develop strongly the brand but also to reinforce significantly its profitability.”
Mayhoola first took control of the brand, then under the Valentino Fashion Group, in 2012. But there has long been speculation that the Middle Eastern group was looking to sell Valentino, with Kering consistently mentioned as a possible buyer.
Giancarlo Giammetti, the longtime business partner of founder Valentino Garavani, said he was “thrilled” by Thursday’s news.
“I have an immense admiration for the work of the Pinault family in the luxury world, with the extraordinary brands they own,” he said.
“We have also been extremely satisfied for how Mayhoola did turn the brand that Mr. Valentino and I founded in 1960 into what it is today,” Giammetti added. “The collaboration between Kering and Valentino holds great potential, and I look forward to seeing how it unfolds in the future.”
A market source dismissed the idea, rumored in the past, that Mayhoola would ultimately divest from its luxury and fashion investments, which also include Balmain and Pal Zileri. “On the contrary, this deal proves its commitment and is in line with the fund’s long-term strategy, since it means Mayhoola could be investing in Kering in time,” the source said.
Jean-Marc Duplaix, Kering’s deputy CEO in charge of operations and finance, clarified that Balmain was not part of the deal “and it’s not contemplated at this stage that there will be something around Balmain going forward.”
A Milan-based financial source said the agreement with Kering “is in the sign of continuity, underscoring the goal to further grow Valentino over the next years, under the lead of Venturini.”
The deal was touted as “a great operation, reflecting an increasing need for companies to bulk up and join forces.”
Kering has been under pressure to make a transformational acquisition that would put it on a more equal footing with rival LVMH Moët Hennessy Louis Vuitton and make it less reliant on Gucci, which accounted for 67 percent of the group’s operating profit last year.
Activist investment firm Bluebell Capital Partners, which was recently reported to have a stake in Kering, had been touting a potential merger with Compagnie Financière Richemont, according to a source with knowledge of the matter, who asked not to be named for confidentiality reasons.
Johann Rupert, Richemont’s controlling shareholder, previously quashed speculation of a combination with Kering. “Everybody urged us to do that a year ago, and two years ago. And we said no,” Rupert said in May.
But after years of failed acquisitions, Kering is finally back on the M&A scene.
Signaling its ambitions in beauty, the French group last month purchased high-end niche fragrance house Creed in a deal reportedly valued at 3.5 billion euros. Pinault said the brand had revenues of around 250 million euros in 2022, with a very high EBITDA margin.
“We have strong growth opportunities for Creed,” he said, noting the brand has very limited exposure to the Asia Pacific region, little to no presence in travel retail, and room to expand its women’s lines.
In turn, Creed’s existing network will allow Kering to build the distribution capabilities for its fledgling beauty division.
The Creed deal came after Kering was in the chase to acquire Tom Ford International, which eventually was bought by that company’s existing beauty licensee, The Estée Lauder Cos. Inc., for $2.3 billion.
On Thursday, Pinault emphasized that in addition to growing its brand portfolio, Kering was fully committed to turning around Gucci, thanks to the new leadership structure unveiled last week.
As part of the reorganization, group managing director Jean-François Palus will take over as president and CEO of Gucci for a transitional period. WWD was the first to report Bizzarri’s departure. His last day at Gucci will be on Sept. 23, after the brand’s spring 2024 show in Milan, the first by new creative director Sabato De Sarno.
Francesca Bellettini, president and CEO of Yves Saint Laurent since 2013, was appointed Kering’s deputy CEO, in charge of brand development. All brand CEOs will report to her, and she will be responsible for steering the group houses in their next stages of growth.
“Saint Laurent has been the most consistent growth story in the group,” said Pinault, adding that Bellettini would remain in charge of that brand.
Duplaix was promoted from chief financial officer, a role he has held since 2012, to deputy CEO. Meanwhile, former Chanel global CEO Maureen Chiquet joined the Kering board.
Pinault said he opted for a seasoned insider at the helm of Gucci in order not to waste time.
“The top priority is to restore the momentum of the top line of Gucci going forward through the relaunch of the aesthetic of Gucci,” he said. “I wanted to be very efficient, very pragmatic. I don’t have years in front of me, I want to put Gucci back on track.”
De Sarno, who has been operational since May, has visited China and is currently in the U.S., he reported. Meanwhile, Palus and Bellettini are ensuring the new look to be unveiled in September will be amplified immediately across all product lines, with the aim of boosting sales before the new products arrive in stores.
Kering will begin the search for a permanent Gucci CEO in September or October, and is open to candidates from outside the luxury sector, Pinault said.
“For me, all the options are open,” he said. “The CEOs in our industry that are already experienced at that size are very, very few, so it’s also a good reason to open to the external world.”
Among the areas for improvement he identified were product quality and supply chain agility. Nonetheless, Pinault was confident that Gucci will reach its medium-term revenue target of 15 billion euros, versus 10.5 billion euros in 2022.
Meanwhile, Kering’s results continue to lag its peers. The group said net profit fell 10 percent to 1.8 billion euros in the first half of 2023 versus the same period last year, as solid growth in Asia was offset by a drop in U.S. sales.
Sales at Gucci totaled 2.51 billion euros in the three months to June 30, up 1 percent on a like-for-like basis, in line with the first quarter. That was below a consensus of analyst estimates, which called for a 4 percent increase in comparable sales at the maker of Jackie 1961 handbags and horsebit loafers, according to a consensus compiled by Bloomberg.
Organic sales at Saint Laurent were up 7 percent in the second quarter, Bottega Veneta gained 3 percent, and the “other houses” division – which groups brands including Balenciaga, Alexander McQueen and Boucheron – posted a 1 percent drop.
By comparison, organic sales at LVMH’s key fashion and leather goods division rose 21 percent year-over-year in the second quarter, reflecting the resilience of its marquee brands Louis Vuitton and Dior.
Kering said retail sales, including e-commerce, were down 23 percent in North America, while the Asia Pacific region gained 22 percent and Japan 26 percent. Western Europe was up 4 percent, and the rest of the world saw a 5 percent progression.
Group revenues in the three months to June 30 rose 2 percent year-on-year to 10.14 billion euros, representing an increase of 2 percent in like-for-like terms. This compared with a 1 percent organic sales increase in the first quarter, and was below the consensus forecast for a 4 percent sales rise in the second quarter.
Recurring operating income fell 3 percent to 2.74 billion euros, yielding an operating margin of 27 percent, down from 28.4 percent in the same period last year.
The Kering results come on the heels of figures from Richemont showing sales at constant exchange rates rose 19 percent in the April-to-June period, fueled by a strong rebound among Chinese tourists and locals.
Moncler said comparable sales were up 26 percent in the second quarter, mainly thanks to an improvement in Asia, while revenues at LVMH rose 17 percent in organic terms during the period. Hermès International is the next big luxury player scheduled to report second-quarter results, on Friday.
Rothschild & Co. acted as adviser of Mayhoola and Centerview Partners advised Kering.
A crucial system of ocean currents is heading for a collapse that ‘would affect every person on the planet’
A vital system of ocean currents could collapse within a few decades if the world continues to pump out planet-heating pollution, scientists are warning – an event that would be catastrophic for global weather and “affect every person on the planet.”
A new study published Tuesday in the journal Nature, found that the Atlantic Meridional Overturning Current – of which the Gulf Stream is a part – could collapse around the middle of the century, or even as early as 2025.
Scientists uninvolved with this study told CNN the exact tipping point for the critical system is uncertain, and that measurements of the currents have so far showed little trend or change. But they agreed these results are alarming and provide new evidence that the tipping point could occur sooner than previously thought.
The AMOC is a complex tangle of currents that works like a giant global conveyor belt. It transports warm water from the tropics toward the North Atlantic, where the water cools, becomes saltier and sinks deep into the ocean, before spreading southwards.
It plays a crucial role in the climate system, helping regulate global weather patterns. Its collapse would have enormous implications, including much more extreme winters and sea level rises affecting parts of Europe and the US, and a shifting of the monsoon in the tropics.
For years, scientists have been warning of its instability as the climate crisis accelerates, threatening to upset the balance of temperature and salinity on which the strength of these currents depend.
As the oceans heat up and ice melts, more freshwater flows into the ocean and reduces the water’s density, making it less able to sink. When waters become too fresh, too warm or both, the conveyor belt stops.
It has happened before. More than 12,000 years ago, rapid glacier melt caused the AMOC to shut down, leading to huge Northern Hemisphere temperature fluctuations of 10 to 15 degrees Celsius (18 to 27 Fahrenheit) within a decade.
A shutdown “would affect every person on the planet – it’s that big and important,” said Peter de Menocal, the president of the Woods Hole Oceanographic Institution, who was not involved in the study.
A 2019 report by the UN’s Intergovernmental Panel on Climate Change predicted that the AMOC would weaken over this century, but that its full collapse before 2100 was unlikely.
This new study comes to a much more alarming conclusion.
As the AMOC has only been continuously monitored since 2004, the study authors looked to a much larger dataset, and one which could show how the currents behaved in a period without human-caused climate change.
“We needed to go back in time,” said Peter Ditlevsen, a professor of climate physics at the University of Copenhagen and one of the authors of the report. The scientists analyzed sea surface temperatures in the North Atlantic in an area south of Greenland over a period of 150 years between 1870 and 2020.
This part of the ocean is warmed by the water transported north from the tropics by the AMOC, Ditlevsen said, “so if it cools, it’s because the AMOC is weakening.” The authors then subtracted the impacts of human-caused global warming on the water temperature to understand how the currents were changing.
They found “early warning signals” of critical changes in the AMOC, which led them to predict “with high confidence” that it could shut down or collapse as early as 2025 and no later than 2095. The likeliest point of collapse is somewhere between 2039 and 2070, Ditlevsen said.
“It’s really scary,” he told CNN. “This is not something you would lightly put into papers,” he said, adding, “we’re very confident that this is a robust result.”
De Menocal said the study results were “both surprising and alarming.”
It’s been clear for a while that the AMOC will weaken in the coming decades, he told CNN. In 2021, a study found the AMOC was showing signs of instability due to climate change.
But until now, we haven’t had a time frame.
The new study “provides a novel analysis that focuses on when the AMOC tipping point will occur,” de Menocal said, and the study’s prediction the collapse will occur around 2050 “is alarmingly soon given the globally disruptive impact of such an event.” Although, he added, it is important to note that there is no observational evidence yet that the AMOC is collapsing.
Stefan Rahmstorf, professor of physics of the oceans at the University of Potsdam in Germany, who was also not involved in the study, said the research helps bolster previous research.
“There is still large uncertainty where the tipping point of the AMOC is, but the new study adds to the evidence that it is much closer than we thought just a few years ago,” he told CNN. “The scientific evidence now is that we can’t even rule out crossing a tipping point already in the next decade or two.”
The report calls for fast and effective measures to cut planet-heating pollution to zero, to reduce global temperatures and slow melting in the Arctic.
“The key point of this study is that we don’t have much time at all to do this,” de Menocal said. “And the stakes just got higher.”
Warning of a forthcoming collapse of the Atlantic meridional overturning circulation
Abstract
The Atlantic meridional overturning circulation (AMOC) is a major tipping element in the climate system and a future collapse would have severe impacts on the climate in the North Atlantic region. In recent years weakening in circulation has been reported, but assessments by the Intergovernmental Panel on Climate Change (IPCC), based on the Climate Model Intercomparison Project (CMIP) model simulations suggest that a full collapse is unlikely within the 21st century. Tipping to an undesired state in the climate is, however, a growing concern with increasing greenhouse gas concentrations. Predictions based on observations rely on detecting early-warning signals, primarily an increase in variance (loss of resilience) and increased autocorrelation (critical slowing down), which have recently been reported for the AMOC. Here we provide statistical significance and data-driven estimators for the time of tipping. We estimate a collapse of the AMOC to occur around mid-century under the current scenario of future emissions.
Introduction
A forthcoming collapse of the Atlantic meridional overturning circulation (AMOC) is a major concern as it is one of the most important tipping elements in Earth’s climate system1,2,3. In recent years, model studies and paleoclimatic reconstructions indicate that the strongest abrupt climate fluctuations, the Dansgaard-Oeschger events4, are connected to the bimodal nature of the AMOC5,6. Numerous climate model studies show a hysteresis behavior, where changing a control parameter, typically the freshwater input into the Northern Atlantic, makes the AMOC bifurcate through a set of co-dimension one saddle-node bifurcations7,8,9. State-of-the-art Earth-system models can reproduce such a scenario, but the inter-model spread is large and the critical threshold is poorly constrained10,11. Based on the CMIP5 generation of models, the AR6 IPCC report quotes a collapse in the 21st century to be very unlikely (medium confidence)12. Among CMIP6 models, there is a larger spread in the AMOC response to warming scenarios, thus an increased uncertainty in the assessment of a future collapse13. There are, however, model biases toward overestimated stability of the AMOC, both from tuning to the historic climate record14, poor representation of the deep water formation15, salinity and glacial runoff16.
When complex systems, such as the overturning circulation, undergo critical transitions by changing a control parameter λ through a critical value λc, a structural change in the dynamics happens. The previously statistically stable state ceases to exist and the system moves to a different statistically stable state. The system undergoes a bifurcation, which for λ sufficiently close to λc can happen in a limited number of ways rather independent from the details in the governing dynamics17. Besides a decline of the AMOC before the critical transition, there are early-warning signals (EWSs), statistical quantities, which also change before the tipping happens. These are critical slowing down (increased autocorrelation) and, from the Fluctuation-Dissipation Theorem, increased variance in the signal18,19,20. The latter is also termed “loss of resilience”, especially in the context of ecological collapse21. The two EWSs are statistical equilibrium concepts. Thus, using them as actual predictors of a forthcoming transition relies on the assumption of quasi-stationary dynamics.
The AMOC has only been monitored continuously since 2004 through combined measurements from moored instruments, induced electrical currents in submarine cables and satellite surface measurements22. Over the period 2004–2012, a decline in the AMOC has been observed, but longer records are necessary to assess the significance. For that, careful fingerprinting techniques have been applied to longer records of sea surface temperature (SST), which, backed by a survey of a large ensemble of climate model simulations, have found the SST in the Subpolar gyre (SG) region of the North Atlantic (area marked with a black contour in Fig. 1a) to contain an optimal fingerprint of the strength of the AMOC23,24,25.
Figure 1b shows the SG and the GM SSTs obtained from the Hadley Centre Sea Ice and Sea Surface Temperature data set (HadISST)26. Figure 1c shows the SG anomaly, and Fig. 1d shows the GM anomaly with a clear global warming trend in the last half of the record. The AMOC fingerprint for the period 1870–2020 is shown in Fig. 1e. This is the basis for the analysis. It has been reported11,27 that this and similar AMOC indices show significant trends in the mean, the variance and the autocorrelation, indicating early warning of a shutdown of the AMOC. However, a trend in the EWSs within a limited period of observation could be a random fluctuation within steady-state statistics. Thus, for a robust assessment of the shutdown, it is necessary to establish a statistical confidence level for the change above the natural fluctuations. This is not easily done given only one, the observed realization of the approach to the transition. Here we establish such a measure of the confidence for the variance and autocorrelation and demonstrate that variance is the more reliable of the two. A further contribution is an estimator of not only whether a transition is approaching but also the time when the critical transition is expected to occur. The strategy is to infer the evolution of the AMOC solely on observed changes in mean, variance and autocorrelation. The typical choice of control parameter is the flux of freshwater into the North Atlantic. River runoff, Greenland ice melt and export from the Arctic Ocean are not well constrained28; thus, we do not assume the control parameter known. Boers27 assumes the global mean temperature T to represent the control parameter. Although T has increased since ~1920 (Fig. 1d), the increase is not quite linear with time. All we assume here is that the AMOC is in an equilibrium state prior to a change toward the transition. The simplest uninformed assumption is that the change is sufficiently slow and that the control parameter approaches the (unknown) critical value linearly with time. This assumption is confirmed by a close fit of the estimated model to the observed AMOC fingerprint. Although we make no explicit assumptions, the primary driver of climate change, the logarithm of the atmospheric CO2 concentration, does, in fact, increase close to linearly with time in the industrial period29. Our results are robust without making specific assumptions regarding the driver of the AMOC.
In this work, we show that a transition of the AMOC is most likely to occur around 2025-2095 (95% confidence interval).
Results
Modeling and detecting the critical transition
Denote the observed AMOC fingerprint by x(t) (Fig. 1e). We model it by a stochastic process Xt, which, depending on a control parameter λ < 0, is at risk of undergoing a critical transition through a saddle-node bifurcation for λ = λc = 0. The system is initially in a statistically stable state, i.e., it follows some stationary distribution with constant λ = λ0. We are uninformed about the dynamics governing the evolution of Xt but can assume effective dynamics, which, with λ sufficiently close to the critical value λc = 0, can be described by the stochastic differential equation (SDE):
dXt=−(A(Xt−m)2+λ)dt+σdBt,
(1)
where m=μ−|λ|/A−−−−−√
and μ is the stable fixed point of the drift, A is a time scale parameter, Bt is a Brownian motion and σ2 scales the variance. Disregarding the noise, this is the normal form of the co-dimension one saddle-node bifurcation17 (see “Methods”). The square-root dependence of the stable state: μ−m∼λc−λ−−−−−√
is the main signature of a saddle-node bifurcation. It is observed for the AMOC shutdown in ocean-only models as well as in coupled models, see Fig. 2, in strong support of Eq. (1) for the AMOC.
At time t0, λ(t) begins to change linearly toward λc = λ(tc) = 0:
λ(t)=λ0(1−Θ[t−t0](t−t0)/τr),
(2)
where Θ[t] is the Heaviside function and τr = tc − t0 > 0 is the ramping time up to time tc, where the transition eventually will occur. Time tc is denoted the tipping time; however, an actual tipping can happen earlier than tc due to a noise-induced tipping. As the transition is approached, the risk of noise-induced tipping (n-tipping) prior to tc is increasing and, at some point, making the EWSs irrelevant for predicting the tipping. The probability for n-tipping can, in the small noise limit, be calculated in closed form, P(t,λ)=1−exp(−t/τn(λ))
, with mean waiting time τn(λ)=(π/|λ|−−√)exp(8|λ|3/2/3σ2)
(see “Methods”).
The mean and variance are calculated from the observations as the control parameter λ(t) is possibly changing. These EWSs are inherently equilibrium concepts and statistical; thus, a time window, Tw, of a certain size is required for a reliable estimate. As the transition is approached, the differences between the EWSs and the pre-ramping values of the variance and autocorrelation (baseline) increase; thus, a shorter window Tw is required for detecting a difference. Conversely, close to the transition critical slowing down decreases the number of independent points within a window, thus calling for a larger window for reliable detection. Within a short enough window, [t − Tw/2, t + Tw/2], we may assume λ(t) to be constant and the noise small enough so that the process (1) for given λ is well approximated by a linear SDE, the Ornstein–Uhlenbeck process30. A Taylor expansion around the fixed point μ(λ) yields the approximation
dXt≈−α(λ)(Xt−μ(λ))dt+σdBt
(3)
where μ(λ)=m+|λ|/A−−−−−√
and α(λ)=2A|λ|−−−−√
is the inverse correlation time. For fixed λ, the process is stationary, with mean μ, variance γ2 = σ2/2α and one-lag autocorrelation ρ=exp(−αΔt)
with step size Δt = 1 month. As λ(t) increases, α decreases, and thus variance and autocorrelation increase. From μ, γ2 and ρ the parameters of Eq. (1) are determined: α=−logρ/Δt
, σ2 = 2αγ2, A = α/2(μ − m) and λ=(σ2/4γ2)2/A
. Closed form estimators for μ, γ2 and ρ are obtained from the observed time series within a running window by maximum likelihood estimation (MLE) (Supplementary text S1, see also ref. 31).
The uncertainty is expressed through the variances of the estimators γ^2
and ρ^
obtained from the observations within a time window Tw. The hats indicate that they are estimators and thus stochastic variables with variances around the true values. Detection of an EWS at some chosen confidence level q (such as 95 or 99%) requires one of the estimates γ^2
or ρ^
for a given window to be statistically different from the baseline values γ^20
or ρ^0
, which depend on the window size as well as how different the EWSs are from their baseline values.
Time scales in early-warning signals
The detection of a forthcoming transition using statistical measures involves several time scales. The primary internal time scale is the autocorrelation time, tac, in the steady state. The ramping time τr over which the control parameter changes from the steady state value to the critical value sets an external time scale. For given α(λ) and q-percentile, the required time window Tw(q, α) to detect a change from baseline in EWSs at the given confidence level q is given in the closed form in the next section (Eq. (7) for variance and Eq. (8) for autocorrelation). The approach to the collapse and the involved time scales are schematically summarized in Fig. 3, while they are calculated in Fig. 4a, where the required window size Tw at the 95% confidence level is plotted as a function of λ for the variance (red curve) and autocorrelation (yellow curve). These are plotted together with the mean waiting time for n-tipping, tnoise, (blue curve). With Tw = 50 years, increased variance can only be detected after the time when λ(t) ≈ −1.2 (crossing of red and red-dashed curves). At that time, a window of approximately 75 years is required to detect an increase in autocorrelation, making variance the better EWS of the two. When λ ≈ −0.4, the mean waiting time for n-tipping is smaller than the data window size. Thus, the increased variance can be used as a reliable EWS in the range −1.2 < λ(t) < −0.4, indicated by the green band. How timely an early warning this is depends on the speed at which λ(t) is changing from λ0 to λc, i.e., the ramping time τr. A set of 1000 realizations has been simulated with λ0 = −2.82 and τr = 140 years, indicated by the time labels on top of Fig. 4a. Ten of these realizations are shown in Fig. 4b on top of the stable and unstable branches of fixed points of model (1) (the bifurcation diagram). Figure 4c (d) shows the variance (autocorrelation) calculated from the realizations within a running 50-year window (shown in Fig. 4c). The solid black line is the baseline value for λ = λ0, while the solid blue line is the increasing value for λ = λ(t). The calculated 95% confidence level for the measurement of the EWS within the running window is shown by the dashed black and blue lines, respectively. The corresponding light blue curves are obtained numerically from the 1000 realizations. The green band in Fig. 4c corresponds to the green band in Fig. 4a and shows where early warning is possible in this case.
Statistics of early-warning signals
The variances of the estimators are approximately (see Supplementary text S1).
Var(γ^2)≈2(γ2)2αTw=σ42α3Tw;Var(ρ^)≈2αΔt2Tw,
(4)
where Tw = nΔt is the observation window.
The question is then how large Tw needs to be to detect a statistically significant increase compared to the estimated baseline values γ^20
and ρ^0
. For a given estimate γ^2
, the estimated difference from the baseline variance is
Δγ2=γ^2−γ^20=γ^20(α^0/α^−1),
(5)
and the estimated difference from the baseline autocorrelation is
Δρ=ρ^−ρ^0=ρ^0(e(α^0−α^)Δt−1)≈ρ^0(α^0−α^)Δt.
(6)
Since the two EWSs, γ^2
and ρ^
, are treated on an equal footing, in the following, we let ψ^
denote either of the estimators (given explicitly in Supplementary text S1, Eqs. (S5) or (S6)). The standard error is s(ψ^)=Var(ψ^)1/2
(Eq. (4)) and Δ^
denotes either of the two estimated differences (5) or (6). The null hypothesis is that λ = λ0, or equivalently α = α0. The null distribution of ψ^
is assumed to be Gaussian (confirmed by simulations). A quantile q from the standard Gaussian distribution expresses the acceptable uncertainty in measuring the statistical quantity ψ. We thus get that Δ^<qs(ψ^)
at the q-confidence level (95%, 99% or similar) under the null hypothesis. To detect an EWS at the q-confidence level based on measuring ψ at time t, we require that Δ^(t)>q(s(ψ^(t))+s(ψ^0))
, which, solved for Tw gives for variance:
Tw>2q2(α^(t)/α^0−−√+α^0/α^(t)−−−−√α^0−α^(t))2,
(7)
and for autocorrelation,
Tw>2q2(α^0−−√+α^(t)−−−−√α^0−α^(t))2ρ^−20.
(8)
Substituting α0=2A|λ0|−−−−−√
and α(t)=2A|λ(t)|−−−−−−√
provides the time window Tw needed to detect an EWS at time t with large probability. Eqs. (7) and (8) are illustrated in Fig. 4a (red and yellow curves), where it is seen that detecting a significant increase in variance requires a shorter data window than detecting a significant increase in autocorrelation. Two times s(ψ^(t))
around the mean of the ramped variance and two times s(ψ^0)
around baseline values are illustrated in Fig. 4c, d (dashed lines). Once a trace leaves the baseline confidence interval, a statistically significant change is detected, and when the two dashed lines cross, 95% of the traces have detected an EWS (Eqs. (7) and (8)).
Predicting a forthcoming collapse of the AMOC
The AMOC fingerprint shown in Fig. 1e (replotted in Fig. 5a) shows an increased variance, γ2, and autocorrelation, ρ, plotted in Fig. 5b, c as functions of the mid-point of a 50-year running window, i.e., the EWS obtained in 2020 is assigned to the year 1995. The estimates leave the confidence band of the baseline values (pink area) around the year 1970. This is not the estimate of t0, which happened earlier and is still to be estimated; it is the year where EWSs are statistically different from baseline values. The estimates after 1970 stay consistently above the upper limit of the confidence interval and show an increasing trend, and we thus conclude that the system is moving toward the tipping point with high probability.
To estimate the tipping time once it has been established that the variance and autocorrelation are increasing, we use two independent methods to check the robustness of our results: (1) Moment-based estimator that uses the variance and autocorrelation estimates within the running windows. (2) Approximate MLE directly on model (1)-(2) with no running window. The advantage of the first method is that it has less model assumptions; however, it is sensitive to the choice of window size. The advantage of the second method is that it uses the information in the data more efficiently given model (1)-(2) is approximately correct, it has no need for a running window and does not assume stationarity after time t0. In general, MLE is statistically the preferred method of choice, giving the most accurate results with the lowest estimation variance.
The first method, the moment estimator of the tipping time obtains, within the running window, the parameters α(t) (Fig. 5d) and σ2 (Fig. 5e) of the linearized dynamics, Eq. (3), and thus also γ2(t). Within the running window, the data are detrended before estimation by subtracting a linear regression fit in order not to falsely inflate the variance estimates caused by deviations from stationarity. Then we obtain Aλ(t) from σ2 and γ2(t) (Fig. 5f) using that Aλ(t)=(σ2/4γ2(t))2
. This is consistent with a linear ramping of λ(t) beginning from a constant level λ0 at a time t0. By sweeping t0 from 1910 to 1950 and Tw from 45 to 65 years, we obtain Aλ0 and τr from the least square error fit to the data. This shows a single minimum at t0 = 1924 and Tw = 55 years (Fig. 6e). Setting t0 = 1924, we obtain tc from a linear fit (regressing λ on t) from the crossing of the x-axis (λc = 0). This is shown in Fig. 5f (red line). This yields −Aλ0 = 2.34 year−2 and τr = 133 years. Thus, the tipping time is estimated to be in the year 2057, shown in Fig. 5f. Since we have only obtained the combined quantity Aλ=(σ2/4γ2)2
, we still need to determine A and m in Eq. (1). We do that from the best linear fit to the mean level μ=m+|λ|/A−−−−−√
observing that μ=m+A|λ|−−−−√(1/A)=m+(σ2/4γ2)(1/A)
. The estimates are shown by the red curves in Fig. 5a–f. The red dot in Fig. 5a is the tipping point, and the dashed line in Fig. 5b is the asymptote for the variance. With the parameter values completely determined, the confidence levels are calculated: The two-standard error levels around the baseline values of the EWS are shown by purple bands in Fig. 5b, c. Thus, both EWSs show increases beyond the two-standard error level from 1970 and onward.
The second method, the approximate MLE of the tipping time, is applied to model (1)–(2). The likelihood function is the product of transition densities between consecutive observations. However, the likelihood is not explicitly known for this model, and we therefore approximate the transition densities. From the data before time t0, approximation (3) is used, where exact MLEs are available (Supplementary text S1). This provides estimates of the parameters λ0, m as a function of parameter A, as well as the variance parameter σ2. To estimate A and τr, the observations after time t0 are used. After time t0, the linear approximation (3) is no longer valid because the dynamics are approaching the bifurcation point, and the non-linear dynamics will be increasingly dominating. The likelihood function is the product of transition densities, which we approximate with a numerical scheme, the Strang splitting, which has shown to have desirable statistical properties for highly non-linear models, where other schemes, such as the Euler–Maruyama approximation is too inaccurate32 (Supplementary text S2). Using t0 = 1924, the optimal fit is the same as the moment method, tc = 2057, with a 95% confidence interval 2025–2095.
Confidence intervals for the estimate of the tipping time are obtained by bootstrap. The likelihood approach provides asymptotic confidence intervals; however, these assume that the likelihood is the true likelihood. To incorporate also the uncertainty due to the data generating mechanism (1) not being equal to the Ornstein–Uhlenbeck process (3) used in the likelihood, we chose to construct parametric bootstrap confidence intervals. This was obtained by simulating 1000 trajectories from the original model with the estimated parameters and repeating the estimation procedure on each data set. Empirical confidence intervals were then extracted from the 1000 parameter estimates. These were indeed larger than the asymptotic confidence intervals provided by the likelihood approach, however, not by much. Histograms of the bootstrapped estimates are shown in Fig. 6a–d. The histogram in Fig. 6a is the tipping year, repeated in yellow in 5f.
The mean of the bootstrapped estimates of the tipping time is 〈tc〉 = 2050, and the 95% confidence interval is 2025–2095. The small discrepancy in the mean is probably due to the approximate model used for estimation being different from the data-generating model (1), confirming that the linear model still provides valid estimates even if the true dynamics are unknown. To test the goodness-of-fit, normal residuals (see “Methods”) were calculated for the data. These are plotted in Fig. 6f as a quantile-quantile plot. If the model is correct, the points fall close to a straight line. The model is seen to fit the data well, further supporting the obtained estimates.
Discussion
We have provided a robust statistical analysis to quantify the uncertainty in observed EWSs for a forthcoming critical transition. The confidence depends on how rapidly the system is approaching the tipping point. With this, the significance of the observed EWSs for the AMOC has been established. This is a stronger result than just observing a significant trend in the EWS by, say, Kendall’s τ test27,33. Here we calculate when the EWS are significantly above the natural variations. Furthermore, we have provided a method to not only determine whether a critical transition will happen but also an estimate of when it will happen. We predict with high confidence the tipping to happen as soon as mid-century (2025–2095 is a 95% confidence range). These results are under the assumption that the model is approximately correct, and we, of course, cannot rule out that other mechanisms are at play, and thus, the uncertainty is larger. However, we have reduced the analysis to have as few and sound assumptions as possible, and given the importance of the AMOC for the climate system, we ought not to ignore such clear indicators of an imminent collapse.
The hysteresis simulations gathered in the model intercomparison34 are equilibrium runs, for which a prediction of a future collapse does obviously not apply. Likewise, for the simulations specified in the CMIP6 experiment. It could though be relevant to evaluate our method on state-of-the-art climate model simulations with linearly ramped external forcing and different ramping speeds in order to obtain the model-specific confidence in early prediction of the collapse judged solely from the EWSs.
Though we have established firm statistical methods to evaluate the confidence in the observed EWS, we can at present not rule out the possibility that a collapse will only be partial and not lead to a full collapse of the AMOC as suggested by some models: Note in Fig. 2, the “MPM” in the top panel and the “MOM hor” in the bottom panel both seem to show only partial tipping prior to the tipping to the complete shutdown of the AMOC. This result is also found in a more recent ocean model35. Furthermore, a high speed of ramping, i.e., a high speed at which the critical value of the control parameter is approached, could also increase the probability of tipping36. This scenario is the case of rate-induced tipping37. Even with these reservations, this is indeed a worrisome result, which should call for fast and effective measures to reduce global greenhouse gas emissions in order to avoid the steady change of the control parameter toward the collapse of the AMOC (i.e., reduce temperature increase and freshwater input through ice melting into the North Atlantic region). As a collapse of the AMOC has strong societal implications38, it is important to monitor the flow and EWS from direct measurements39,40,41.
Methods
To obtain the AMOC fingerprint, two steps are required: The seasonal cycle in the SST is governed by the surface radiation independent from the circulation and thus removed by considering the monthly anomalies, where the mean over the period of recording of the month is removed. Second, there is an ongoing positive linear trend in the SST related to global warming, which is also not related to circulation. This is compensated for by subtracting 2 × the global mean (GM) SST anomaly (small seasonal cycle removed). This differs slightly from ref. 23, where 1 × the GM SST was subtracted. The “translation” from the proxy SST temperature and AMOC flow is 0.26 SV/K [ref. 23, Fig. 3]. Here we have taken into account that the warming is not globally homogeneous: The warming in the SG region is larger than the global mean due to polar amplification. The way we have estimated this effect is by comparing the proxy with the AMOC estimates covering the period 1957–2004 from the so-called MOCz reported in the review by ref. 42. This shows a drop of 3 SV in that period. Minimizing the difference between the proxy SSTSG-β SSTGM and this more direct measurement with respect to β, we get β = 1.95 ≈ 2 rather than β = 1 used by ref. 23. The factor 2 is thus the optimal value for the polar amplification43 obtained by calibrating to recent direct measurements42. The original and our calibrated proxies are shown in Fig. 7.
To check the robustness with respect to the AMOC fingerprint record, we repeated the analysis, subtracting 1x and 3x GM SST from the SG SST. Subtracting 3x GM SST only changes estimate and the confidence intervals by a few years, whereas subtracting 1x GM SST delays the tipping with 25 years, but the overall trend and conclusions do not change. The results are given in Table 1. In the reanalysis, we fixed t0 = 1924.
Estimator of the tipping time and model control
The AMOC fingerprint is assumed to be observations from a process Xt given as a solution to Eqs. (1) and (2), and we wish to estimate the parameters θ = (A, m, λ0, τr, σ) from observations (x0, x1, …, xn) before time t0 and observations (y0, y1, …, yn) after time t0. These equations cannot be explicitly solved, and the exact distribution of Xt is not explicitly known. A standard way to solve this is to approximate the transition density by a Gaussian distribution obtained by the Euler–Maruyama scheme. However, the estimators obtained from the Euler–Maruyama pseudo-likelihood are known to be biased, especially in non-linear models32. We estimate by a two-step procedure, approximating the stationary distribution before time t0 by an Ornstein–Uhlenbeck process, of which exact maximum likelihood estimators are available (see Supplementary text S1), and using Strang splitting for the non-stationary and non-linear part after time t0, using methods proposed in ref. 32, see Supplementary text S2 for details.
To test the model fit, uniform residuals, ui, i = 1, …, n were calculated for the AMOC data using the estimated parameters from the MLE method as follows. The model assumes that observation xi follows some distribution function Fi,θ^
for the estimated parameter values θ^
. If this is true, then ui=Fi,θ^(xi)
is uniformly distributed on (0, 1). Transforming these residuals back to a standard normal distribution provides standard normally distributed residuals if the model is true. Thus, a normal quantile-quantile plot reveals the model fit. The points should fall close to a straight line. The reason for making the detour around the uniform residuals is twofold. First, since the data is not stationary, each observation follows its own distribution, and residuals cannot be directly combined. Second, since the model is stochastic, standard residuals are not well-defined, and observations should be evaluated according to their entire distribution, not only the distance to the mean.
Noise-induced tipping
The drift term in Eq. (1) is the negative gradient of a potential, f(x, λ) = − ∂xV(x, λ) = − (A(x−m)2 + λ) with V(x, λ) = A(x−m)3/3 + (x − m)λ. For λ < 0, the drift has two fixed points, m±|λ|/A−−−−−√
. The point m+|λ|/A−−−−−√
is a local minimum of the potential V(x, λ) and is stable, whereas m−|λ|/A−−−−−√
is a local maximum and unstable. The system thus has two basins of attraction separated by m−|λ|/A−−−−−√
, with a drift toward either m+|λ|/A−−−−−√
or −∞ dependent on whether Xt>m−|λ|/A−−−−−√
or Xt<m−|λ|/A−−−−−√
. We denote the two basins of attraction, the normal and the tipped state, respectively. When λ = 0, the normal state disappears, and the system undergoes a bifurcation and Xt will be drawn toward −∞.
Due to the noise, the process can escape into the tipped state by crossing over the potential barrier Δ(λ)=V(m−|λ|/A−−−−−√,λ)−V(m+|λ|/A−−−−−√,λ)=4|λ|3/2/3A1/2
. Assume Xt to be close to m+|λ|/A−−−−−√
at some time t, i.e., in the normal state. The escape time will asymptotically (for σ → 0) follow an exponential distribution such that
P(t,λ)=1−exp(−t/τn(λ))
(9)
where P(t, λ) is the probability of observing an escape time shorter than t for a given value of λ. The mean noise-induced escape time τn(λ) is44,45:
τn(λ)=2πexp(2Δ(λ)/σ2)V′′(m+|λ|/A−−−−−√,λ)|V′′(m−|λ|/A−−−−−√,λ)|−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√=(π/A|λ|−−−−√)exp(8|λ|3/2/3A1/2σ2).
(10)
Assume that the rate of change of λ(t) follows Eq. (2), then for τr < τn(λ), the waiting time for a random crossing is so long that a crossing will not happen before a bifurcation-induced transition happens (b-tipping). If τr > τn(λ), a noise-induced tipping is expected before the bifurcation point is reached. Since τn(λ) decreases with increasing λ, at some point, the two time scales will end up matching.
Normal form of the saddle-node bifurcation
Consider the general dynamical equation
dxdt=f(x,λ),
(11)
where x is a variable and λ is a (fixed) parameter. A point x0 with f(x0, λ) = 0 is a fixed point or steady state. A fixed point is stable/unstable if ∂xf(x,λ)x=x0
is negative/positive; thus, the fixed point is attracting/repelling. If f(x, λ) is not a linear function of x, multiple steady states may exist. A saddle-node bifurcation occurs when changing the control parameter λ through a critical value λc a stable and an unstable fixed point merge and disappear. The situation is shown in the figure, where the blue surface is f(x, λ), while the gray (null-) plane is f(x, λ) = 0. For a constant value of λ, the dynamics is determined by the black curve. The fixed points are determined by the intersection with the null plane (green); the point in the front is the stable fixed point, while the further point is the unstable fixed point. When changing λ toward λc = 0, the two fixed points merge at the saddle-node bifurcation (m, λc) (green). The normal form of the saddle node is obtained by expanding f(x, λ) to the lowest order around the point (m, λc), noting that f(m, λc) = 0, ∂xf(x,λ)(x,λ)=(m,λc)=0
and ∂λf(x,λ)(x,λ)=(m,λc)<0
(see Fig. 8):
f(x,λ)≈12∂xxf(x,λ)(x,λ)=(m,λc)×(x−m)2+∂λf(x,λ)(x,λ)=(m,λc)×(λ−λc)=−A(x−m)2−λ~,
(12)
where A=−12∂xxf(x,λ)(x,λ)=(m,λc)
and λ~=−∂λf(x,λ)(x,λ)=(m,λc)×(λ−λc)
. This is the normal form for the saddle-node bifurcation. Thus, close to the bifurcation point, the stable steady state is
x0=m+−λ~/A−−−−−√.
In order to see that this is indeed the case for the AMOC transition also in comprehensive climate models, Fig. 2 is adapted from the model intercomparison study34. The steady state curves obtained are from simulations, with a very slowly changing control parameter (freshwater forcing). The top panel shows ocean-only models, while the bottom panel shows atmosphere-ocean models. The curves are, even away from the transition, surprisingly well fitted by Eq. (13). Note that for some models, the transition happens before the critical point, as should be expected from noise-induced transitions. Note also that the data has been smoothed such that increasing variance close to the transition is not visible. This observation strongly supports the assumption of a saddle-node bifurcation, while it also shows that (m, λc) (black dots) are quite different between models, thus calling for reliable determination from observations.
Warning of a forthcoming collapse of the Atlantic meridional overturning circulation
Abstract
The Atlantic meridional overturning circulation (AMOC) is a major tipping element in the climate system and a future collapse would have severe impacts on the climate in the North Atlantic region. In recent years weakening in circulation has been reported, but assessments by the Intergovernmental Panel on Climate Change (IPCC), based on the Climate Model Intercomparison Project (CMIP) model simulations suggest that a full collapse is unlikely within the 21st century. Tipping to an undesired state in the climate is, however, a growing concern with increasing greenhouse gas concentrations. Predictions based on observations rely on detecting early-warning signals, primarily an increase in variance (loss of resilience) and increased autocorrelation (critical slowing down), which have recently been reported for the AMOC. Here we provide statistical significance and data-driven estimators for the time of tipping. We estimate a collapse of the AMOC to occur around mid-century under the current scenario of future emissions.
Introduction
A forthcoming collapse of the Atlantic meridional overturning circulation (AMOC) is a major concern as it is one of the most important tipping elements in Earth’s climate system1,2,3. In recent years, model studies and paleoclimatic reconstructions indicate that the strongest abrupt climate fluctuations, the Dansgaard-Oeschger events4, are connected to the bimodal nature of the AMOC5,6. Numerous climate model studies show a hysteresis behavior, where changing a control parameter, typically the freshwater input into the Northern Atlantic, makes the AMOC bifurcate through a set of co-dimension one saddle-node bifurcations7,8,9. State-of-the-art Earth-system models can reproduce such a scenario, but the inter-model spread is large and the critical threshold is poorly constrained10,11. Based on the CMIP5 generation of models, the AR6 IPCC report quotes a collapse in the 21st century to be very unlikely (medium confidence)12. Among CMIP6 models, there is a larger spread in the AMOC response to warming scenarios, thus an increased uncertainty in the assessment of a future collapse13. There are, however, model biases toward overestimated stability of the AMOC, both from tuning to the historic climate record14, poor representation of the deep water formation15, salinity and glacial runoff16.
When complex systems, such as the overturning circulation, undergo critical transitions by changing a control parameter λ through a critical value λc, a structural change in the dynamics happens. The previously statistically stable state ceases to exist and the system moves to a different statistically stable state. The system undergoes a bifurcation, which for λ sufficiently close to λc can happen in a limited number of ways rather independent from the details in the governing dynamics17. Besides a decline of the AMOC before the critical transition, there are early-warning signals (EWSs), statistical quantities, which also change before the tipping happens. These are critical slowing down (increased autocorrelation) and, from the Fluctuation-Dissipation Theorem, increased variance in the signal18,19,20. The latter is also termed “loss of resilience”, especially in the context of ecological collapse21. The two EWSs are statistical equilibrium concepts. Thus, using them as actual predictors of a forthcoming transition relies on the assumption of quasi-stationary dynamics.
The AMOC has only been monitored continuously since 2004 through combined measurements from moored instruments, induced electrical currents in submarine cables and satellite surface measurements22. Over the period 2004–2012, a decline in the AMOC has been observed, but longer records are necessary to assess the significance. For that, careful fingerprinting techniques have been applied to longer records of sea surface temperature (SST), which, backed by a survey of a large ensemble of climate model simulations, have found the SST in the Subpolar gyre (SG) region of the North Atlantic (area marked with a black contour in Fig. 1a) to contain an optimal fingerprint of the strength of the AMOC23,24,25.
Figure 1b shows the SG and the GM SSTs obtained from the Hadley Centre Sea Ice and Sea Surface Temperature data set (HadISST)26. Figure 1c shows the SG anomaly, and Fig. 1d shows the GM anomaly with a clear global warming trend in the last half of the record. The AMOC fingerprint for the period 1870–2020 is shown in Fig. 1e. This is the basis for the analysis. It has been reported11,27 that this and similar AMOC indices show significant trends in the mean, the variance and the autocorrelation, indicating early warning of a shutdown of the AMOC. However, a trend in the EWSs within a limited period of observation could be a random fluctuation within steady-state statistics. Thus, for a robust assessment of the shutdown, it is necessary to establish a statistical confidence level for the change above the natural fluctuations. This is not easily done given only one, the observed realization of the approach to the transition. Here we establish such a measure of the confidence for the variance and autocorrelation and demonstrate that variance is the more reliable of the two. A further contribution is an estimator of not only whether a transition is approaching but also the time when the critical transition is expected to occur. The strategy is to infer the evolution of the AMOC solely on observed changes in mean, variance and autocorrelation. The typical choice of control parameter is the flux of freshwater into the North Atlantic. River runoff, Greenland ice melt and export from the Arctic Ocean are not well constrained28; thus, we do not assume the control parameter known. Boers27 assumes the global mean temperature T to represent the control parameter. Although T has increased since ~1920 (Fig. 1d), the increase is not quite linear with time. All we assume here is that the AMOC is in an equilibrium state prior to a change toward the transition. The simplest uninformed assumption is that the change is sufficiently slow and that the control parameter approaches the (unknown) critical value linearly with time. This assumption is confirmed by a close fit of the estimated model to the observed AMOC fingerprint. Although we make no explicit assumptions, the primary driver of climate change, the logarithm of the atmospheric CO2 concentration, does, in fact, increase close to linearly with time in the industrial period29. Our results are robust without making specific assumptions regarding the driver of the AMOC.
In this work, we show that a transition of the AMOC is most likely to occur around 2025-2095 (95% confidence interval).
Results
Modeling and detecting the critical transition
Denote the observed AMOC fingerprint by x(t) (Fig. 1e). We model it by a stochastic process Xt, which, depending on a control parameter λ < 0, is at risk of undergoing a critical transition through a saddle-node bifurcation for λ = λc = 0. The system is initially in a statistically stable state, i.e., it follows some stationary distribution with constant λ = λ0. We are uninformed about the dynamics governing the evolution of Xt but can assume effective dynamics, which, with λ sufficiently close to the critical value λc = 0, can be described by the stochastic differential equation (SDE):
dXt=−(A(Xt−m)2+λ)dt+σdBt,
(1)
where m=μ−|λ|/A−−−−−√
and μ is the stable fixed point of the drift, A is a time scale parameter, Bt is a Brownian motion and σ2 scales the variance. Disregarding the noise, this is the normal form of the co-dimension one saddle-node bifurcation17 (see “Methods”). The square-root dependence of the stable state: μ−m∼λc−λ−−−−−√
is the main signature of a saddle-node bifurcation. It is observed for the AMOC shutdown in ocean-only models as well as in coupled models, see Fig. 2, in strong support of Eq. (1) for the AMOC.
At time t0, λ(t) begins to change linearly toward λc = λ(tc) = 0:
λ(t)=λ0(1−Θ[t−t0](t−t0)/τr),
(2)
where Θ[t] is the Heaviside function and τr = tc − t0 > 0 is the ramping time up to time tc, where the transition eventually will occur. Time tc is denoted the tipping time; however, an actual tipping can happen earlier than tc due to a noise-induced tipping. As the transition is approached, the risk of noise-induced tipping (n-tipping) prior to tc is increasing and, at some point, making the EWSs irrelevant for predicting the tipping. The probability for n-tipping can, in the small noise limit, be calculated in closed form, P(t,λ)=1−exp(−t/τn(λ))
, with mean waiting time τn(λ)=(π/|λ|−−√)exp(8|λ|3/2/3σ2)
(see “Methods”).
The mean and variance are calculated from the observations as the control parameter λ(t) is possibly changing. These EWSs are inherently equilibrium concepts and statistical; thus, a time window, Tw, of a certain size is required for a reliable estimate. As the transition is approached, the differences between the EWSs and the pre-ramping values of the variance and autocorrelation (baseline) increase; thus, a shorter window Tw is required for detecting a difference. Conversely, close to the transition critical slowing down decreases the number of independent points within a window, thus calling for a larger window for reliable detection. Within a short enough window, [t − Tw/2, t + Tw/2], we may assume λ(t) to be constant and the noise small enough so that the process (1) for given λ is well approximated by a linear SDE, the Ornstein–Uhlenbeck process30. A Taylor expansion around the fixed point μ(λ) yields the approximation
dXt≈−α(λ)(Xt−μ(λ))dt+σdBt
(3)
where μ(λ)=m+|λ|/A−−−−−√
and α(λ)=2A|λ|−−−−√
is the inverse correlation time. For fixed λ, the process is stationary, with mean μ, variance γ2 = σ2/2α and one-lag autocorrelation ρ=exp(−αΔt)
with step size Δt = 1 month. As λ(t) increases, α decreases, and thus variance and autocorrelation increase. From μ, γ2 and ρ the parameters of Eq. (1) are determined: α=−logρ/Δt
, σ2 = 2αγ2, A = α/2(μ − m) and λ=(σ2/4γ2)2/A
. Closed form estimators for μ, γ2 and ρ are obtained from the observed time series within a running window by maximum likelihood estimation (MLE) (Supplementary text S1, see also ref. 31).
The uncertainty is expressed through the variances of the estimators γ^2
and ρ^
obtained from the observations within a time window Tw. The hats indicate that they are estimators and thus stochastic variables with variances around the true values. Detection of an EWS at some chosen confidence level q (such as 95 or 99%) requires one of the estimates γ^2
or ρ^
for a given window to be statistically different from the baseline values γ^20
or ρ^0
, which depend on the window size as well as how different the EWSs are from their baseline values.
Time scales in early-warning signals
The detection of a forthcoming transition using statistical measures involves several time scales. The primary internal time scale is the autocorrelation time, tac, in the steady state. The ramping time τr over which the control parameter changes from the steady state value to the critical value sets an external time scale. For given α(λ) and q-percentile, the required time window Tw(q, α) to detect a change from baseline in EWSs at the given confidence level q is given in the closed form in the next section (Eq. (7) for variance and Eq. (8) for autocorrelation). The approach to the collapse and the involved time scales are schematically summarized in Fig. 3, while they are calculated in Fig. 4a, where the required window size Tw at the 95% confidence level is plotted as a function of λ for the variance (red curve) and autocorrelation (yellow curve). These are plotted together with the mean waiting time for n-tipping, tnoise, (blue curve). With Tw = 50 years, increased variance can only be detected after the time when λ(t) ≈ −1.2 (crossing of red and red-dashed curves). At that time, a window of approximately 75 years is required to detect an increase in autocorrelation, making variance the better EWS of the two. When λ ≈ −0.4, the mean waiting time for n-tipping is smaller than the data window size. Thus, the increased variance can be used as a reliable EWS in the range −1.2 < λ(t) < −0.4, indicated by the green band. How timely an early warning this is depends on the speed at which λ(t) is changing from λ0 to λc, i.e., the ramping time τr. A set of 1000 realizations has been simulated with λ0 = −2.82 and τr = 140 years, indicated by the time labels on top of Fig. 4a. Ten of these realizations are shown in Fig. 4b on top of the stable and unstable branches of fixed points of model (1) (the bifurcation diagram). Figure 4c (d) shows the variance (autocorrelation) calculated from the realizations within a running 50-year window (shown in Fig. 4c). The solid black line is the baseline value for λ = λ0, while the solid blue line is the increasing value for λ = λ(t). The calculated 95% confidence level for the measurement of the EWS within the running window is shown by the dashed black and blue lines, respectively. The corresponding light blue curves are obtained numerically from the 1000 realizations. The green band in Fig. 4c corresponds to the green band in Fig. 4a and shows where early warning is possible in this case.
Statistics of early-warning signals
The variances of the estimators are approximately (see Supplementary text S1).
Var(γ^2)≈2(γ2)2αTw=σ42α3Tw;Var(ρ^)≈2αΔt2Tw,
(4)
where Tw = nΔt is the observation window.
The question is then how large Tw needs to be to detect a statistically significant increase compared to the estimated baseline values γ^20
and ρ^0
. For a given estimate γ^2
, the estimated difference from the baseline variance is
Δγ2=γ^2−γ^20=γ^20(α^0/α^−1),
(5)
and the estimated difference from the baseline autocorrelation is
Δρ=ρ^−ρ^0=ρ^0(e(α^0−α^)Δt−1)≈ρ^0(α^0−α^)Δt.
(6)
Since the two EWSs, γ^2
and ρ^
, are treated on an equal footing, in the following, we let ψ^
denote either of the estimators (given explicitly in Supplementary text S1, Eqs. (S5) or (S6)). The standard error is s(ψ^)=Var(ψ^)1/2
(Eq. (4)) and Δ^
denotes either of the two estimated differences (5) or (6). The null hypothesis is that λ = λ0, or equivalently α = α0. The null distribution of ψ^
is assumed to be Gaussian (confirmed by simulations). A quantile q from the standard Gaussian distribution expresses the acceptable uncertainty in measuring the statistical quantity ψ. We thus get that Δ^<qs(ψ^)
at the q-confidence level (95%, 99% or similar) under the null hypothesis. To detect an EWS at the q-confidence level based on measuring ψ at time t, we require that Δ^(t)>q(s(ψ^(t))+s(ψ^0))
, which, solved for Tw gives for variance:
Tw>2q2(α^(t)/α^0−−√+α^0/α^(t)−−−−√α^0−α^(t))2,
(7)
and for autocorrelation,
Tw>2q2(α^0−−√+α^(t)−−−−√α^0−α^(t))2ρ^−20.
(8)
Substituting α0=2A|λ0|−−−−−√
and α(t)=2A|λ(t)|−−−−−−√
provides the time window Tw needed to detect an EWS at time t with large probability. Eqs. (7) and (8) are illustrated in Fig. 4a (red and yellow curves), where it is seen that detecting a significant increase in variance requires a shorter data window than detecting a significant increase in autocorrelation. Two times s(ψ^(t))
around the mean of the ramped variance and two times s(ψ^0)
around baseline values are illustrated in Fig. 4c, d (dashed lines). Once a trace leaves the baseline confidence interval, a statistically significant change is detected, and when the two dashed lines cross, 95% of the traces have detected an EWS (Eqs. (7) and (8)).
Predicting a forthcoming collapse of the AMOC
The AMOC fingerprint shown in Fig. 1e (replotted in Fig. 5a) shows an increased variance, γ2, and autocorrelation, ρ, plotted in Fig. 5b, c as functions of the mid-point of a 50-year running window, i.e., the EWS obtained in 2020 is assigned to the year 1995. The estimates leave the confidence band of the baseline values (pink area) around the year 1970. This is not the estimate of t0, which happened earlier and is still to be estimated; it is the year where EWSs are statistically different from baseline values. The estimates after 1970 stay consistently above the upper limit of the confidence interval and show an increasing trend, and we thus conclude that the system is moving toward the tipping point with high probability.
To estimate the tipping time once it has been established that the variance and autocorrelation are increasing, we use two independent methods to check the robustness of our results: (1) Moment-based estimator that uses the variance and autocorrelation estimates within the running windows. (2) Approximate MLE directly on model (1)-(2) with no running window. The advantage of the first method is that it has less model assumptions; however, it is sensitive to the choice of window size. The advantage of the second method is that it uses the information in the data more efficiently given model (1)-(2) is approximately correct, it has no need for a running window and does not assume stationarity after time t0. In general, MLE is statistically the preferred method of choice, giving the most accurate results with the lowest estimation variance.
The first method, the moment estimator of the tipping time obtains, within the running window, the parameters α(t) (Fig. 5d) and σ2 (Fig. 5e) of the linearized dynamics, Eq. (3), and thus also γ2(t). Within the running window, the data are detrended before estimation by subtracting a linear regression fit in order not to falsely inflate the variance estimates caused by deviations from stationarity. Then we obtain Aλ(t) from σ2 and γ2(t) (Fig. 5f) using that Aλ(t)=(σ2/4γ2(t))2
. This is consistent with a linear ramping of λ(t) beginning from a constant level λ0 at a time t0. By sweeping t0 from 1910 to 1950 and Tw from 45 to 65 years, we obtain Aλ0 and τr from the least square error fit to the data. This shows a single minimum at t0 = 1924 and Tw = 55 years (Fig. 6e). Setting t0 = 1924, we obtain tc from a linear fit (regressing λ on t) from the crossing of the x-axis (λc = 0). This is shown in Fig. 5f (red line). This yields −Aλ0 = 2.34 year−2 and τr = 133 years. Thus, the tipping time is estimated to be in the year 2057, shown in Fig. 5f. Since we have only obtained the combined quantity Aλ=(σ2/4γ2)2
, we still need to determine A and m in Eq. (1). We do that from the best linear fit to the mean level μ=m+|λ|/A−−−−−√
observing that μ=m+A|λ|−−−−√(1/A)=m+(σ2/4γ2)(1/A)
. The estimates are shown by the red curves in Fig. 5a–f. The red dot in Fig. 5a is the tipping point, and the dashed line in Fig. 5b is the asymptote for the variance. With the parameter values completely determined, the confidence levels are calculated: The two-standard error levels around the baseline values of the EWS are shown by purple bands in Fig. 5b, c. Thus, both EWSs show increases beyond the two-standard error level from 1970 and onward.
The second method, the approximate MLE of the tipping time, is applied to model (1)–(2). The likelihood function is the product of transition densities between consecutive observations. However, the likelihood is not explicitly known for this model, and we therefore approximate the transition densities. From the data before time t0, approximation (3) is used, where exact MLEs are available (Supplementary text S1). This provides estimates of the parameters λ0, m as a function of parameter A, as well as the variance parameter σ2. To estimate A and τr, the observations after time t0 are used. After time t0, the linear approximation (3) is no longer valid because the dynamics are approaching the bifurcation point, and the non-linear dynamics will be increasingly dominating. The likelihood function is the product of transition densities, which we approximate with a numerical scheme, the Strang splitting, which has shown to have desirable statistical properties for highly non-linear models, where other schemes, such as the Euler–Maruyama approximation is too inaccurate32 (Supplementary text S2). Using t0 = 1924, the optimal fit is the same as the moment method, tc = 2057, with a 95% confidence interval 2025–2095.
Confidence intervals for the estimate of the tipping time are obtained by bootstrap. The likelihood approach provides asymptotic confidence intervals; however, these assume that the likelihood is the true likelihood. To incorporate also the uncertainty due to the data generating mechanism (1) not being equal to the Ornstein–Uhlenbeck process (3) used in the likelihood, we chose to construct parametric bootstrap confidence intervals. This was obtained by simulating 1000 trajectories from the original model with the estimated parameters and repeating the estimation procedure on each data set. Empirical confidence intervals were then extracted from the 1000 parameter estimates. These were indeed larger than the asymptotic confidence intervals provided by the likelihood approach, however, not by much. Histograms of the bootstrapped estimates are shown in Fig. 6a–d. The histogram in Fig. 6a is the tipping year, repeated in yellow in 5f.
The mean of the bootstrapped estimates of the tipping time is 〈tc〉 = 2050, and the 95% confidence interval is 2025–2095. The small discrepancy in the mean is probably due to the approximate model used for estimation being different from the data-generating model (1), confirming that the linear model still provides valid estimates even if the true dynamics are unknown. To test the goodness-of-fit, normal residuals (see “Methods”) were calculated for the data. These are plotted in Fig. 6f as a quantile-quantile plot. If the model is correct, the points fall close to a straight line. The model is seen to fit the data well, further supporting the obtained estimates.
Discussion
We have provided a robust statistical analysis to quantify the uncertainty in observed EWSs for a forthcoming critical transition. The confidence depends on how rapidly the system is approaching the tipping point. With this, the significance of the observed EWSs for the AMOC has been established. This is a stronger result than just observing a significant trend in the EWS by, say, Kendall’s τ test27,33. Here we calculate when the EWS are significantly above the natural variations. Furthermore, we have provided a method to not only determine whether a critical transition will happen but also an estimate of when it will happen. We predict with high confidence the tipping to happen as soon as mid-century (2025–2095 is a 95% confidence range). These results are under the assumption that the model is approximately correct, and we, of course, cannot rule out that other mechanisms are at play, and thus, the uncertainty is larger. However, we have reduced the analysis to have as few and sound assumptions as possible, and given the importance of the AMOC for the climate system, we ought not to ignore such clear indicators of an imminent collapse.
The hysteresis simulations gathered in the model intercomparison34 are equilibrium runs, for which a prediction of a future collapse does obviously not apply. Likewise, for the simulations specified in the CMIP6 experiment. It could though be relevant to evaluate our method on state-of-the-art climate model simulations with linearly ramped external forcing and different ramping speeds in order to obtain the model-specific confidence in early prediction of the collapse judged solely from the EWSs.
Though we have established firm statistical methods to evaluate the confidence in the observed EWS, we can at present not rule out the possibility that a collapse will only be partial and not lead to a full collapse of the AMOC as suggested by some models: Note in Fig. 2, the “MPM” in the top panel and the “MOM hor” in the bottom panel both seem to show only partial tipping prior to the tipping to the complete shutdown of the AMOC. This result is also found in a more recent ocean model35. Furthermore, a high speed of ramping, i.e., a high speed at which the critical value of the control parameter is approached, could also increase the probability of tipping36. This scenario is the case of rate-induced tipping37. Even with these reservations, this is indeed a worrisome result, which should call for fast and effective measures to reduce global greenhouse gas emissions in order to avoid the steady change of the control parameter toward the collapse of the AMOC (i.e., reduce temperature increase and freshwater input through ice melting into the North Atlantic region). As a collapse of the AMOC has strong societal implications38, it is important to monitor the flow and EWS from direct measurements39,40,41.
Methods
To obtain the AMOC fingerprint, two steps are required: The seasonal cycle in the SST is governed by the surface radiation independent from the circulation and thus removed by considering the monthly anomalies, where the mean over the period of recording of the month is removed. Second, there is an ongoing positive linear trend in the SST related to global warming, which is also not related to circulation. This is compensated for by subtracting 2 × the global mean (GM) SST anomaly (small seasonal cycle removed). This differs slightly from ref. 23, where 1 × the GM SST was subtracted. The “translation” from the proxy SST temperature and AMOC flow is 0.26 SV/K [ref. 23, Fig. 3]. Here we have taken into account that the warming is not globally homogeneous: The warming in the SG region is larger than the global mean due to polar amplification. The way we have estimated this effect is by comparing the proxy with the AMOC estimates covering the period 1957–2004 from the so-called MOCz reported in the review by ref. 42. This shows a drop of 3 SV in that period. Minimizing the difference between the proxy SSTSG-β SSTGM and this more direct measurement with respect to β, we get β = 1.95 ≈ 2 rather than β = 1 used by ref. 23. The factor 2 is thus the optimal value for the polar amplification43 obtained by calibrating to recent direct measurements42. The original and our calibrated proxies are shown in Fig. 7.
To check the robustness with respect to the AMOC fingerprint record, we repeated the analysis, subtracting 1x and 3x GM SST from the SG SST. Subtracting 3x GM SST only changes estimate and the confidence intervals by a few years, whereas subtracting 1x GM SST delays the tipping with 25 years, but the overall trend and conclusions do not change. The results are given in Table 1. In the reanalysis, we fixed t0 = 1924.
Estimator of the tipping time and model control
The AMOC fingerprint is assumed to be observations from a process Xt given as a solution to Eqs. (1) and (2), and we wish to estimate the parameters θ = (A, m, λ0, τr, σ) from observations (x0, x1, …, xn) before time t0 and observations (y0, y1, …, yn) after time t0. These equations cannot be explicitly solved, and the exact distribution of Xt is not explicitly known. A standard way to solve this is to approximate the transition density by a Gaussian distribution obtained by the Euler–Maruyama scheme. However, the estimators obtained from the Euler–Maruyama pseudo-likelihood are known to be biased, especially in non-linear models32. We estimate by a two-step procedure, approximating the stationary distribution before time t0 by an Ornstein–Uhlenbeck process, of which exact maximum likelihood estimators are available (see Supplementary text S1), and using Strang splitting for the non-stationary and non-linear part after time t0, using methods proposed in ref. 32, see Supplementary text S2 for details.
To test the model fit, uniform residuals, ui, i = 1, …, n were calculated for the AMOC data using the estimated parameters from the MLE method as follows. The model assumes that observation xi follows some distribution function Fi,θ^
for the estimated parameter values θ^
. If this is true, then ui=Fi,θ^(xi)
is uniformly distributed on (0, 1). Transforming these residuals back to a standard normal distribution provides standard normally distributed residuals if the model is true. Thus, a normal quantile-quantile plot reveals the model fit. The points should fall close to a straight line. The reason for making the detour around the uniform residuals is twofold. First, since the data is not stationary, each observation follows its own distribution, and residuals cannot be directly combined. Second, since the model is stochastic, standard residuals are not well-defined, and observations should be evaluated according to their entire distribution, not only the distance to the mean.
Noise-induced tipping
The drift term in Eq. (1) is the negative gradient of a potential, f(x, λ) = − ∂xV(x, λ) = − (A(x−m)2 + λ) with V(x, λ) = A(x−m)3/3 + (x − m)λ. For λ < 0, the drift has two fixed points, m±|λ|/A−−−−−√
. The point m+|λ|/A−−−−−√
is a local minimum of the potential V(x, λ) and is stable, whereas m−|λ|/A−−−−−√
is a local maximum and unstable. The system thus has two basins of attraction separated by m−|λ|/A−−−−−√
, with a drift toward either m+|λ|/A−−−−−√
or −∞ dependent on whether Xt>m−|λ|/A−−−−−√
or Xt<m−|λ|/A−−−−−√
. We denote the two basins of attraction, the normal and the tipped state, respectively. When λ = 0, the normal state disappears, and the system undergoes a bifurcation and Xt will be drawn toward −∞.
Due to the noise, the process can escape into the tipped state by crossing over the potential barrier Δ(λ)=V(m−|λ|/A−−−−−√,λ)−V(m+|λ|/A−−−−−√,λ)=4|λ|3/2/3A1/2
. Assume Xt to be close to m+|λ|/A−−−−−√
at some time t, i.e., in the normal state. The escape time will asymptotically (for σ → 0) follow an exponential distribution such that
P(t,λ)=1−exp(−t/τn(λ))
(9)
where P(t, λ) is the probability of observing an escape time shorter than t for a given value of λ. The mean noise-induced escape time τn(λ) is44,45:
τn(λ)=2πexp(2Δ(λ)/σ2)V′′(m+|λ|/A−−−−−√,λ)|V′′(m−|λ|/A−−−−−√,λ)|−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√=(π/A|λ|−−−−√)exp(8|λ|3/2/3A1/2σ2).
(10)
Assume that the rate of change of λ(t) follows Eq. (2), then for τr < τn(λ), the waiting time for a random crossing is so long that a crossing will not happen before a bifurcation-induced transition happens (b-tipping). If τr > τn(λ), a noise-induced tipping is expected before the bifurcation point is reached. Since τn(λ) decreases with increasing λ, at some point, the two time scales will end up matching.
Normal form of the saddle-node bifurcation
Consider the general dynamical equation
dxdt=f(x,λ),
(11)
where x is a variable and λ is a (fixed) parameter. A point x0 with f(x0, λ) = 0 is a fixed point or steady state. A fixed point is stable/unstable if ∂xf(x,λ)x=x0
is negative/positive; thus, the fixed point is attracting/repelling. If f(x, λ) is not a linear function of x, multiple steady states may exist. A saddle-node bifurcation occurs when changing the control parameter λ through a critical value λc a stable and an unstable fixed point merge and disappear. The situation is shown in the figure, where the blue surface is f(x, λ), while the gray (null-) plane is f(x, λ) = 0. For a constant value of λ, the dynamics is determined by the black curve. The fixed points are determined by the intersection with the null plane (green); the point in the front is the stable fixed point, while the further point is the unstable fixed point. When changing λ toward λc = 0, the two fixed points merge at the saddle-node bifurcation (m, λc) (green). The normal form of the saddle node is obtained by expanding f(x, λ) to the lowest order around the point (m, λc), noting that f(m, λc) = 0, ∂xf(x,λ)(x,λ)=(m,λc)=0
and ∂λf(x,λ)(x,λ)=(m,λc)<0
(see Fig. 8):
f(x,λ)≈12∂xxf(x,λ)(x,λ)=(m,λc)×(x−m)2+∂λf(x,λ)(x,λ)=(m,λc)×(λ−λc)=−A(x−m)2−λ~,
(12)
where A=−12∂xxf(x,λ)(x,λ)=(m,λc)
and λ~=−∂λf(x,λ)(x,λ)=(m,λc)×(λ−λc)
. This is the normal form for the saddle-node bifurcation. Thus, close to the bifurcation point, the stable steady state is
x0=m+−λ~/A−−−−−√.
In order to see that this is indeed the case for the AMOC transition also in comprehensive climate models, Fig. 2 is adapted from the model intercomparison study34. The steady state curves obtained are from simulations, with a very slowly changing control parameter (freshwater forcing). The top panel shows ocean-only models, while the bottom panel shows atmosphere-ocean models. The curves are, even away from the transition, surprisingly well fitted by Eq. (13). Note that for some models, the transition happens before the critical point, as should be expected from noise-induced transitions. Note also that the data has been smoothed such that increasing variance close to the transition is not visible. This observation strongly supports the assumption of a saddle-node bifurcation, while it also shows that (m, λc) (black dots) are quite different between models, thus calling for reliable determination from observations.
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Mining companies warn of weak commodity price outlook
‘Covid overhangs’ in Chinese economy have sapped demand for metals, says Anglo American chief
Some of the world’s largest mining companies have become increasingly pessimistic about a sustained rally in global commodity prices this year after an insipid recovery in China led to a drop in first-half earnings.
Duncan Wanblad, chief executive of Anglo American, said on Thursday that an “enormous amount of economic pressure” was making it hard to wager that sustained gains in the prices of its key commodities such as iron ore, copper and steelmaking coal would take place by the end of the year.
“We’ve been a little bit surprised at how slow the opening of China has been,” he said, adding that the world’s largest metals-consuming nation was still suffering from “Covid overhangs” that had sapped demand.
In reference to when commodity prices would bottom out and rally, he added that “it’s more likely to be early next year than later this year quite honestly”.
His comments came after the FTSE 100 miner cut dividends by more than half to $700mn following a fall in first-half core earnings of just over 40 per cent to $5.1bn. This occurred off the back of a 19 per cent drop in the price of its commodities and inflationary cost pressures.
On Monday, Beijing vowed to boost consumer spending and revive the economy but fell short of announcing stimulus measures that would help manufacturing rebound, prompting a shortlived bounce in metal prices.
Citigroup analyst Wenyu Yao said that China’s politburo meeting set a “positive tone” yet it was not the sort of “bazooka-style” stimulus required to push up Chinese demand for metals in the second half of the year.
Goldman Sachs analysts said that “the industrial metals complex has been caught for much of this year between disappointing growth momentum in China, a manufacturing slowdown in the west and accelerating supply for several metals”.
But they saw enough of a shift point in China’s rhetoric to merit upgrading the bank’s six-month copper price forecast by 3 per cent to $9,500 per tonne, compared with $8,600 per tonne at present.
Mining companies have been touting the prospective boost to prices from green policies around the world since building wind farms, transmission grids and electric cars requires a lot of metals.
Simon Morris, head of base metals at CRU Group, a business intelligence company, said that a lift to demand from the clean energy transition looked unlikely to materialise as quickly as hoped, while China is unlikely to compensate given a shift in its economic model.
“Will China continue to fuel demand as it has done? No . . . China won’t be as strong a demand story as it has been for 20 years,” he said. “There’s a lot of hyperbole around the energy transition.”
Rio Tinto, the world’s largest producer of iron ore, which is used to make steel, said on Wednesday its main commodities were trading below their 2010 average prices during the first half of this year.
“We saw lower prices, in general, for our commodities, in line with slowing global demand, with the Chinese recovery predominantly led by the service sector,” the company said in its earnings release.
Chief financial officer Peter Cunningham said the future direction of iron ore prices, which are down 10 per cent since January, would depend on what happens to the “extremely soft property market” in China.
The difficult conditions were further confirmed by shares in French mining group Eramet falling 12 per cent on Thursday after it revised down its full-year earnings because of expected weakness in manganese and nickel ore prices.