Weighted Sobolev Spaces

We now give the definition of weighted Sobolev spaces, which will be needed in Chapter 4, as defined in [Del09].

Definition 1.6.1 Forα∈[0,1[andm≥1 the Weighted Sobolev space H^m,α(Ω)is defined as H^m,α(Ω) :={u∈H^m−1(Ω) : |u|²_Hm,α(Ω)= X

|β|=m

kr^αD^βuk²_L2(Ω)<∞}

wherer:=r(x)is the distance fromx∈Ωto the origin.

H^0,α(Ω)is the set of measurable functions such that their semi-norm verifies

|u|²_H0,α(Ω):=kr^αuk²_L2(Ω)<∞.

Definition 1.6.2 The norm onH^m,α(Ω) is defined by

kuk²_Hm,α(Ω)=kuk²_Hm−1(Ω)+|u|²_Hm,α(Ω) if m≥1, kuk²_H0,α(Ω)=|u|²_H0,α(Ω) if m= 1.

Theorem 1.6.1 The spaceH^m,α(Ω)is a Hilbert space.

Proof We give the proof for m= 1, knowing that it is very straightforward to generalise.

Let {fn}_n≥1 ⊂ H^1,α be a Cauchy sequence. Then for all > 0 there existsN such that for all m, n > N we have

kfn−fmk²_H1,α =kfn−fmk²_L2+

2

X

i=1

kr^α∂x_i(fn−fm)k²_L2 < .

We can deduce from this that {fn}n≥1 and {r^α∂x_ifn}n≥1 are Cauchy sequences in L²(Ω). So there existsf, g∈L²(Ω) such that

n→∞lim kfn−fkL² = 0, lim

n→∞kr^α∂xifn−gkL² = 0.

Moreover, from each of these sequences we can extract a subsequence which tends to itsL²(Ω) limit almost everywhere i.e.

∃ {fnk}k≥1such thatf_nk −→f a.e.

∃ {r^α∂x_ifn_k}_k≥1such thatr^α∂x_ifn_k−→g a.e.

Letϕbe a distribution on Ω. We have ˆ

Ω

∂_xif_nkϕ dx=− ˆ

Ω

f_nk∂_xiϕ dx.

Taking the limit and using the fact that lim

k→∞∂_xif_nk= lim

k→∞r^−αr^α∂_xif_nk =r^−α lim

k→∞r^α∂_xif_nk =r^−αg

leads to ˆ

Ω

r^−αgϕ dx=− ˆ

Ω

f ∂x_iϕ dx= ˆ

Ω

∂x_if ϕ dx.

We thus have thatg=r^α∂_xif which allows us to conclude.

Theorem 1.6.2 Let 1< p < _1+α² . We have the following embeddings:

(a) W^1,p(Ω)⊂^c L²(Ω), (b) W^2,p(Ω)⊂^c H¹(Ω), (c) H^0,α(Ω)⊂W^0,p(Ω), (d) H^1,α(Ω)⊂W^1,p(Ω), (e) H^2,α(Ω)⊂W^2,p(Ω).

(f ) H^2,α(Ω)⊂^c H¹(Ω).

Moreover, ifα <1, we have (g) H^2,α(Ω)⊂ C⁰( ¯Ω).

Proof

(a) This is a consequence of the Rellich-Kondrachov Theorem (see for instance [Bre87]).

(b) Letf ∈W^2,p(Ω). This implies that ∇f ∈W^1,p(Ω). By (a) there existsC such that k∇fk_L2(Ω)≤Ck∇fk_W1,p(Ω)

⇔kfk_H1(Ω)≤C



k∇fk_Lp(Ω)+ X

|β|=2

kD^βfk_Lp(Ω)



≤Ckfk_W2,p(Ω).

This shows thatW^2,p(Ω)⊂H¹(Ω). Let {fn}n≥1 be a sequence in W^2,p(Ω). Using (a) and the fact thatW^2,p(Ω)⊂W^1,p(Ω) one can extract a subsequence which is Cauchy inL²(Ω)

kfn_k−fn_lk_L2(Ω)< n whenk, l−→ ∞.

As we have{∇f_nk}_k≥1⊂W^1,p(Ω) we can again extract a Cauchy sequence inL²(Ω) k∇(fn_ki−fn_kj)kL²(Ω)< n wheni, j−→ ∞.

So{∇fn_ki}i≥1 is a Cauchy sequence inH¹(Ω) which implies that the embedding is compact.

(c) Letf ∈H^0,α(Ω). We have

kfkW^0,p(Ω)≤CkfkH^0,α(Ω)⇔ kfkL^p(Ω)≤Ckr^αfkL²(Ω). Let >0. By Theorem 1.5.8, we have that for ¹_p ≥¹₂+ 2¹

α−: kr^αf r^−αkL^p(Ω)=kfkL^p(Ω)≤ kr^αfkL²(Ω)kr^−αk

Lα²−(Ω). The last term can be bounded, thanks to theterm

kr^−αk^α2−

Lα²−(Ω)= ˆ

Ω

r^−2−α:=C <∞.

So we have

kfk_Lp(Ω)≤Ckr^αfk_L2(Ω)

for all ¹_p ≥ ¹₂+ 2¹

α− > ¹₂+ ¹2 α

which allows us to conclude.

(d) Letf ∈H^1,α(Ω) and p < _1+α² . We have to show that there existsCsuch that

kfkW^1,p(Ω)=kfkL^p(Ω)+k∇fkL^p(Ω)≤CkfkH^1,α(Ω)=C(kfkL²(Ω)+kr^α∇fkL²(Ω)).

By definition,f ∈H^1,α(Ω) implies that∇f ∈H^0,α(Ω) so by (c) there existsC1 such that k∇fkL^p(Ω)≤C1kr^α∇fkL²(Ω).

Sincep <2 we can use Corollary 1.5.7 to findC₂ such that kfk_Lp(Ω)≤C2kfk_L2(Ω)

which allows us to conclude.

(e) Letf ∈H^2,α(Ω) and p < _1+α² . We have to show that there existsCsuch that kfk_W2,p(Ω)=kfk_Lp(Ω)+k∇fk_Lp(Ω)+ X

|β|=2

kD^βfk_Lp(Ω)

≤CkfkH^2,α(Ω)=C(kfkL²(Ω)+k∇fkL^p(Ω)+ X

|β|=2

kr^αD^βfkL^p(Ω)).

Sincep <2 we have by Corollary 1.5.7 that there existC1 andC2such that kfkL^p(Ω)≤C₁kfkL²(Ω), k∇fkL^p(Ω)≤C₂k∇fkL²(Ω). Moreover by definition ofH^2,α(Ω) we have

X

|β|=2

kr^αD^βfkL^p(Ω)<∞,

which implies that D^βf ∈ H^0,α(Ω) for every multi-indexβ such that |β| = 2. So by (c) there existsC3such that

kD^βfk_Lp(Ω)≤C3kr^αD^βfk_L2(Ω) ∀|β|= 2, which allows us to conclude.

(f) Using (e) and (b) gives the result.

(g) We apply Theorem 1.5.11 withs= 2,k= 0 to obtain W^2,p(Ω)⊂ C^0,2−2p, 2−2

p ∈R>0\N. Using the fact that

C^0,k⊂ C⁰ ∀k <1 and (e) yields

H^2,α(Ω)⊂W^2,p(Ω)⊂ C^0,2−2p ⊂ C⁰∀p < 2

1 +α, α <1.

Theorem 1.6.3 The weighted Sobolev space H^1,α(Ω) is continuously embedded in L²(∂Ω) for all α < ¹₂ i.e.

H^1,α(Ω),→L²(∂Ω)∀α < 1 2.

Proof Combining Theorem 1.6.2 (d) and Theorem 1.5.9 withN = 2 andq= 2 gives the result.

The following is the Lemma 8.4.1.3 from [Gri85].

Lemma 1.6.4 LetP1(Ω)be the space of the first-order polynomials restricted toΩ. Letf ∈H^2,α(Ω).

Then there exists a constant C such that inf

p∈P1(Ω)kf −pk²_H2,α(Ω)≤C|f|²_H2,α(Ω). Proof

1. We first show that

kvk²_H2,α(Ω)≤C|v|²_H2,α(Ω) ∀v∈P₁(Ω)^⊥, (1.6.1) where P1(Ω)^⊥ is the orthogonal ofP1(Ω) inH^2,α(Ω) i.e.

P1(Ω)^⊥ = (

q=

N

X

i=1

qixⁱ, N∈N:hq, pi_H2,α(Ω)= 0∀p∈P1(Ω) )

, where hf, giH^2,α(Ω):=kf gk²_H1(Ω)+|f g|²_H2,α(Ω).

Showing (1.6.1) is equivalent to showing inf

v∈P1(Ω)^⊥, kvk_H2,α(Ω)=1

C²|v|²_H2,α(Ω)>0. (1.6.2)

We now prove (1.6.2) by contradiction i.e. we suppose that there existsw∈P1(Ω)^⊥ such that inf

v∈P1(Ω)^⊥, kvk_H2,α(Ω)=1

C²|v|²_H2,α(Ω)= 0.

Then there exists a sequence{vn}_n≥1⊂P1(Ω)^⊥ withkvnk_H2,α(Ω)= 1∀nsuch that

n→∞lim C²|vn|H^2,α(Ω)= 0. (1.6.3) By Theorem 1.6.2 (f),H^2,α(Ω) is compactly embedded intoH¹(Ω) so there exists a subsequence {v_nk}_k≥1 which is a Cauchy sequence inH¹(Ω). This and equation (1.6.3) show that {v_nk}_k≥1 is a Cauchy sequence in H^2,α(Ω):

kvn_k−vn_lk²_H2,α(Ω)=kvn_k−vn_lk²_H1(Ω)+ X

|β|=2

kr^αD^β(vn_k−vn_l)k²_L2(Ω).

By Theorem 1.6.1 we know thatH^2,α is a Hilbert space so there existsv∈H^2,α such that vn_k −→v inH^2,α(Ω).

SincekvnkH^2,α(Ω)= 1 andP1(Ω)^⊥ ∀n, we also have (a) kvk_H2,α(Ω)= 1

(b) v∈P1(Ω)^⊥ .

But from (1.6.3) we haveD^βw= 0 for|β|= 2 which impliesw∈P¹(Ω). In conclusion, we have w∈P¹(Ω)∩P¹(Ω)^⊥⇒w= 0

which contradicts (a) so (1.6.2) cannot be true.

2. Let us setv=u−pwhere pis the orthogonal projection ofuontoP¹(Ω). Applying (1.6.1) to v leads to the result.

The following Lemma follows [Gri85].

Lemma 1.6.5 Let Ω⊂R² be a polygonal domain, and let xi i = 1, . . . , n denote the vertices of Ω.

Let α∈[0,1[andf ∈H^2,α(Ω). We denote by Πf the first order interpolating polynomial i.e.

Πf ∈P¹(Ω)andΠf(xi) =f(xi)∀i= 1, . . . , n. (1.6.4) There existsC >0such that

kf−ΠfkH¹(Ω)≤C|f|H^2,α(Ω).

Proof First of all, Theorem 1.6.2 (g) tells us that the definition of Πf in (1.6.4) makes sense. Then, we notice that for everyp∈P¹(Ω) we have

f−Πf = (I−Π)(f−p). (1.6.5)

Since the identity operator and Π are continuous fromH^2,α(Ω) intoH¹(Ω) there exists a constant C such that

kI−Πk_H2,α(Ω)−→H¹(Ω)≤C.

Using (1.6.5) we get

kf−Πfk_H1(Ω)=k(I−Π)(f −p)k_H1(Ω)≤Ckf−pk_H2,α(Ω)

⇔ inf

p∈P¹(Ω)kf−ΠfkH¹(Ω)=kf−ΠfkH¹(Ω)≤C inf

p∈P¹(Ω)kf−pkH^2,α(Ω). We then apply Lemma 1.6.4 to conclude.

Chapter 2

A Very Short Introduction to Numerical Weather Prediction

Moonrise gives similar indications, at the time of full moon: they are less certain when the moon is not full. If the moon looks fiery, it indicates breezy weather for that month, if dusky, wet weather; and, whatever indications the crescent moon gives, are given when it is three days old.

Θεόφραστος(Theophrastus) c. 371 – c. 287 BC,On Weather Signs

In this chapter, we first attempt to give a brief introduction to Numerical Weather Predic-tion (NWP), based mainly on “Invisible in the Storm: The Role of Mathematics in Understanding Weather” by I. Roulstone and J. Norbury ([RN13]). We then give a few notions which are commonly used in NWP nowadays.

2.1 A Historical Introduction to NWP

Forecasting the weather is a question that mankind has been working on for thousands of years. As early as 650 BC, the Babylonians attempted to predict weather changes by studying astronomical and meteorological events. The first revolution in the context of meteorology was due to two inventions credited to Galileo Galileiand his students in the early 1600s: the thermometer and the baro-meter. From that point, scientist started tabulating variations of pressure, temperature and rainfall in different countries. These observations were analysed and regularities or patterns were searched for.

The second revolutionary event in the history of meteorology is due, ironically, to an astronomer:

Edmond Halley. In 1685 (twenty years before correctly predicting the return of the comet that would be named after him), Halley derived what we now call thehydrostatic equation; the equation that describes how pressure decreases with height. Halley used what is calledBoyle’s law to derive this equation. Boyle’s law is anequation of statewhich arose fromRobert Boyle’s discovery in 1660 that at a given temperature, the pressure and volume occupied by a gas are related to one another.

The following year, Halley published the first meteorological chart of the winds over the oceans. He was trying to understand what causes the winds, a question already addressed two thousand years earlier by Aristotle. Halley described the general wind over the oceans as a current of air which is due to the warming effect of the Sun. His explanation was not entirely correct but the drifting motion he mentions is part of what we today call the general circulation of the atmosphere: heat from equatorial regions moves to colder polar regions. Until 1757, Newton’s laws of motion were applied to solid objects but no one knew how to deal with fluids. Then, Leonhard Euler came into the picture and constructed the basis of hydrodynamics. He formulated four equations which describe the way pressure acts to change the motion of a parcel of fluid. The first three describe the motion in the three directions of space. The fourth represents what we call the conservation of mass i.e.

the fact that motion does not create or destroy fluid matter. In the 1850s, William Thomson (Lord

Figure 2.1.1: Meteorological chart,An Historical Account of the Trade Winds, and Monsoons, observ-able in the Seas between and near the Tropicks, with an Attempt to Assign the Phisical Cause of the Said Winds, E. Halley, 1686.

Kelvin) andRudolf Clausiusdevised two principles which are known as the first and second laws of thermodynamics. The first, also known as theconservation of energy, describes the movement of heat energy and the related change in pressure, density and temperature, the total energy remaining constant. The second states that while the total quantity of energy must be conserved in any process, the distribution of that energy changes in an irreversible manner. At that stage, six out of the seven equations that form the basis of our physical model of the atmosphere and oceans were known (see Box 2.1.4). However, using them to predict the weather was not a priority for scientists of that time.

An important year in the history of weather forecasting is the year 1854, just one year into the Crimean War. On November 13 the French and British fleets were almost entirely destroyed by a storm over the Black Sea. This disaster made the headlines and the population demanded an explanation.

At that time, astronomers were predicting dates and times of eclipses and the ebbing and flowing of tides accurately enough for many years in the future. Louis Napol´eon (Napol´eon Bonaparte’s nephew) was the Emperor of the French at that time, and he asked the superstar of astronomy of the moment,Urbain Le Verrier, if the Black Sea Storm would have been predictable. Le Verrier and his colleagues analysed the data from weather reports from locations across Europe for the period 10-16 November (see Figure 2.1.2 for an example synoptic chart). The conclusion of this analysis was that the track of the storm could have been anticipated.

This led to an international call for cooperation, for the establishment of a network of weather stations and the free exchange of atmospheric data for the purposes of weather forecasting. The data collected consisted of the air pressure measured with a barometer and wind speed and direction measured with anemometers and weather vanes. Those measures were transmitted to forecasting centers by telegraph and the forecasters would predict where the weather systems would then move.

Another major event in the history of weather forecasting was reached in 1854 whenRobert FitzRoy was appointed to be the leader of the just created Meteorological Board of Trade in Britain. He thus earned the title of first national weather forecaster. He had gained knowledge about meteorology from his years in the navy and he was the captain of the Beagle in 1831, during its well-known expedition to South America. FitzRoy was the one who selectedCharles Darwinto be the scientist accompanying the Beagle’s expedition. His efforts to standardize all meteorological observations facilitated the exchange of data between the United States and England, and allowed the construction of same-day weather charts in those regions.

The next big achievement was attained byWilliam Ferrel, when he derived the definitive equa-tions of meteorology (see Figure 2.1.4). He was one of the first to realise that the rotation of the Earth dominates the behavior of the weather. This phenomenon is now known as theCoriolis effect, because Gaspard-Gustave de Coriolisderived the mathematical expression for the Coriolis force in 1835. Unfortunately, this was not known by most physicists and the link with meteorology was not made at that time. From 1858 to 1860 Ferrel constructed a new theory of the general circulation of the atmosphere around our planet (see Figure 2.1.3). He was able to give the detailed equations that describe the global weather patterns but he was not able to solve them explicitly.

Figure 2.1.2: Synoptic chart used by Le Verrier to analyse the Black Sea storm in 1854.

Figure 2.1.3: Trade of winds and basic mechanism of cyclone rotation inAn essay on the winds and the currents of the ocean, William Ferrel, 1856.

The Seven Equations that form the basis of weather prediction

The wind is the movement of air parcels with velocity v = (u, v, w), density ρ and mass δm=ρδV whereδV is the volume of the parcel. The Conservation of momentumis represented by the wind equations

Du Dt =F1

ρ , Dv Dt = F2

ρ , Dw Dt =F3

ρ ,

whereF= (F1, F2, F3) includes the pressure gradient, the effects of gravity and the Earth’s rotation, and frictional forces acting on a fluid parcel.

TheEquation of State(or theideal gas law) links the pressurep, the density ρand the absolute temperatureT

p=ρRT, whereR is the ideal gas constant.

TheConservation of Energyis formulated as follows Q=cv

DT Dt − p

ρ² Dρ Dt,

where cv is the constant know as the specific heat at constant volume and Qis the heating rate.

Finally, we have two equations for the Conservation of Mass. The density equation reads

Dρ

Dt =−ρ∇ ·v.

And the equation for water vaporqis Dq Dt =S,

where S is the net supply of water to an air parcel due to condensation and evaporation processes.

Figure 2.1.4: The Seven Equations that form the basis of weather prediction.

Figure 2.1.5: Science et M´ethode, H. Poincar´e, 1903.

The hope for a weather warning service was very high and in 1861, FitzRoy started issuing storm warnings. In 1863, he began forecasting to the public via the newspapers which was a huge success, even though the observations were scarce. The forecasts of the time usually were quite concise; they would for instance announce “rainy and windy” or “clear and cold” weather. However, some scientists were very skeptical as to forecasts, due to the lack of scientific basis. The laws of physics were not taken into account in the forecasting process and more and more people mocked forecasters. They were associated with folklore and were constantly ridiculed. In 1865, FitzRoy committed suicide.

The Board of Trade then decided to end the issuance of forecasts and storm warnings. It took two years and many storms destroying vessels to reinstate those warnings, at the public demand.

Another ten years later, the publication of weather forecasts was reintroduced by the Meteorological Office (the former Board of Trade) in spite of meteorology still being considered as the antithesis of science. It was in this relatively unfriendly context that Vilhelm Bjerknes published a paper in 1904 which revolutionised meteorology by formulating weather prediction as a problem in physics and mathematics. He introduced two necessary steps for finding the solution to the forecasting problem:

1. Diagnostic: A sufficiently accurate knowledge of the state of the atmosphere at the initial time.

2. Prognostic: A sufficiently accurate knowledge of the laws according to which one state of the atmosphere develops from another.

This was the start of modern weather forecasting. Just one year before the paper from Bjerknes was published, an essay byHenri Poincar´eunsettled the old belief that the future can be predetermined by knowing the laws of nature. Poincar´e wrote this essay when studying one of the big challenges of the time: establishing if the solar system would remain stable for all time. The conclusion he reaches can be found in his essay (see Figure 2.1.5): “Prediction (of the situation of the universe) is impossible and fortuitous phenomena exist.” This discovery of unpredictability, which we now call chaos, might have undermined Bjerknes’s visionary conceptualisation of weather prediction. Luckily, accepting the phenomenon of chaos forced mathematicians to realize that new techniques were needed in order to study chaotic systems.

The idea Bjerknes proposed in his 1904 paper was to calculate a forecast at a finite number of points in the atmosphere, but he did not give more detail as to how to solve the seven equations. In fact, he believed that working with the equations directly was impractical, so he and his team developed

Figure 2.1.6: The finite-difference grid used by Richardson to produce the first numerical weather forecast, 1922.

graphical methods based on charts and on the circulation theorem to understand the weather. Another scientist of that time,Lewis Fry Richardson, was inspired by Bjerknes’s work and devised a method for solving the equations. This he did whilst driving ambulances in France during the First World War. He drew a grid over Europe (see Figure 2.1.6) and divided the depth of the atmosphere into five layers. He attempted to forecast the weather during a single day by direct computation of finite differences (see Section 1.1) by hand. It took him two years and the forecast was utterly wrong. It was nonetheless the first attempt to forecast weather by direct computation as opposed to extrapolating from weather patterns, and the birth of numerical weather prediction.

The Second World War then erupted and one of the benefits of it was the accelerated progress in the development of computing machines. The impact of weather during the conflict was undeniable and many weather research programs were initiated using military resources. The first public mul-tipurpose electronic, digital computing machine was the ENIAC (Electronic Numerical Integrator and Computer). It became operational in December 1945. In May 1946, John von Neumann wrote a proposal to the Office of Research and Inventions (Navy Department), asking for support for

“an investigation of the theory of dynamic meteorology in order to make it accessible to high speed, electronic, digital, automatic computing”. It took a couple of years, but in 1950, ENIAC performed a twenty-four-hour forecast for North America. The program was designed by Jule Charney and von Neumann. The computation time for the forecast was of twenty-four hours (the same time it had taken for the events to take place) but the results were very encouraging.

In the 1950sEdward Lorenzwas working on a simplified weather model, using a quite powerful computer for those times. He was looking for periodic patterns in the weather. What he found instead was that a very small error in initial conditions could be responsible for the divergence of the solution.

A couple of years later, Lorenz reportedly said that the flapping of a butterfly’s wings over Brazil might cause a tornado over Texas. This led to the definition of what we now call thebutterfly effect, the fact that small differences in initial conditions produce very large ones in the final phenomenon (see Figure 2.1.7). It was also the beginning of chaos theory. Although butterfly wings preclude the exact forecasting of weather, we may still estimate future climatic trends with reasonable precision.

The approach which is used by most modern weather centers is ensemble forecasting. In 1969 Edward Epsteinpublished a paper in the Journal of Applied Meteorology in which he studied the

Figure 2.1.7: Evolution of two trajectories with starting points which differ very slightly. The second cone indicates the final position of both trajectories, which are not coincident. Source: [WLO].

ensemble of forecasts made from a set of predictive equations with slightly differing initial conditions.

At that time, most countries had their own meteorological institutions. However it soon became clear that an intergovernmental organisation producing medium-range weather forecasts was needed.

TheEuropean Centre for Medium-Range Weather Forecasts(ECMWF) was created in 1975 and the eighteen founding states were: Austria, Belgium, Croatia, Denmark, Finland, France, Ger-many, Greece, Republic of Ireland, Italy, Luxembourg, Netherlands, Norway, Portugal, Spain, Sweden, Switzerland, Turkey and the United Kingdom. Since 2010, four more countries joined the ECMWF:

Iceland, Slovenia, Serbia and Croatia.

The Consortium for Small-scale Modeling (COSMO) was formed in 1998. The COSMO model is a non-hydrostatic limited-area atmospheric model and it is used both for operational and for research applications by the members of the consortium. The members are the following national met-eorological services: Germany (DWD), Switzerland (MeteoSchweiz), Italy (ReMet), Greece (HNMS), Poland (IMGW), Romania (NMA), Russia (RHM) and Israel (IMS). Since the COSMO-Model is a limited-area model, it needs lateral boundary conditions from a driving model. One of the options is to use data from the ECMWF as boundary conditions.

Ensemble forecasting started being used operationally in the 1990s. The ensemble forecast is a probabilistic forecast, obtained by observing several equally probable forecasts. If the individual forecasts vary a lot, the uncertainty about the weather to come is high. However, if the forecasts are similar regarding the occurrence of a particular event then the probability that the event will take place is very close to one. Most operational weather prediction facilities now conduct ensemble forecasts.

The ECMWF started doing so in 1992, and it now runs 51 forecasts with the same starting time and slightly different initial conditions. Ensemble forecasting was made operational in the COSMO model in 2016 with the COSMO-E forecasting system. 21 equally probable forecasts are calculated twice per

The ECMWF started doing so in 1992, and it now runs 51 forecasts with the same starting time and slightly different initial conditions. Ensemble forecasting was made operational in the COSMO model in 2016 with the COSMO-E forecasting system. 21 equally probable forecasts are calculated twice per

Dans le document The Discrete Duality Finite Volume method in the context of weather prediction models (Page 39-51)