The Hovmöller diagram or the trough–ridge diagram, a simple longitude–time diagram, was designed in 1948 by Ernest Hovmöller (1912–2008) to help understand certain features in the dynamics of the atmosphere, in particular the “downstream development” phenomenon. Originally depicting the 500-hPa geopotential, today many other parameters are used, and Hovmöller diagrams have during the last 25 years found a rapidly increasing use in a wide range of atmospheric research.
The Hovmöller diagram, a seemingly simple time–longitude construction, played a decisive role in the early development of computerized weather forecasts.
In June 1991, a few months after I had joined the European Centre for Medium-Range Forecasts (ECMWF), the management decided that the weather charts in the map room should be complemented with daily Hovmöller diagrams. These time–longitude diagrams with longitude along the x axis and time along the y axis, depicting the temporal–spatial variation of a chosen meteorological parameter, were constructed by the Danish–Swedish meteorologist Ernest Hovmöller in 1948 (Fig. 1). The original Hovmöller diagram, using 500-hPa geopotential heights, was intended to highlight not only the positions of the planetary waves but also incidents of downstream development, a successive development and/or amplifications of troughs and ridges (Fig. 2).
At ECMWF a colorful draft version was soon produced, covering 30 rolling days—the last 20 days plus the next 10 days from the operational forecast system. It was then natural to phone up Mr. Hovmöller, who at that time was in retirement outside Stockholm, Sweden, to make him aware that his diagram was still being used. During the conversation he shared details about the story of “his” diagram.
ERNEST HOVMÖLLER’S EARLY CAREER.
Ernest Hovmöller joined the Danish Meteorological Institute (DMI) in 1937 at the age of 25. He became interested in the condition in the upper atmosphere even before he joined the DMI and in 1936 spent 4 months at the leading aerological center in Lindenberg outside Berlin, Germany.
The predominating attitude at DMI in the late 1930s was that aerological measurements, if reliable at all, could hardly be of any practical value. Hovmöller sensed that at the neighboring Swedish Meteorological and Hydrological Institute (SMHI), a more progressive attitude prevailed. His plans to visit the SMHI were cancelled when World War II broke out and Denmark was occupied. When peace came, Hovmöller together with his wife and children made the journey to Sweden. Hovmöller was employed and placed in the expanding aerological section at SMHI in November 1946. Soon another immigrant—Carl Gustaf Rossby—joined the SMHI after 20 years in the United States (Berson 1991; Phillips 1998).
Rossby started a series of lectures (in English) to bring the staff at SMHI up to date on scientific developments in meteorology, in particular dynamic meteorology. Hovmöller perhaps did not understand all of Rossby’s advanced lectures; he was in particular puzzled about something called group velocity, about which Rossby seemed to be keen.
THE NOTION OF GROUP VELOCITY.
Rossby’s interest in group velocity dates back to summer 1944. Then he visited the Scripps Institution of Oceanography in La Jolla, California, where the oceanographers Harald Sverdrup and Walter Munk were conducting war-related secret work on ocean waves. Such waves are dispersive; their phase velocity is proportional to the wavelength: the longer the ocean wave, the faster it moves. For dispersive waves the energy is propagating with the group velocity of the wave system. In the case of ocean waves, their group velocity—and thus the energy—will travel at half the speed of the waves (Fig. 3).
Perhaps it was while listening to the varying strength of the incoming waves slashing on the California beach that it dawned on Rossby that the group velocity concept could also apply to motions in the atmosphere. His well-known wave formula is also dispersive. In its most simple form, it can be written as
where c is the phase speed; U is the average tropospheric zonal current, often represented by the 500-hPa wind; k = 2π/λ is the wavenumber, where λ is the wavelength; and β is the meridional variation of the Coriolis parameter. These waves are dispersive but in an opposite way to ocean waves: the longer-wavelength troughs tend to move slower or even westward relative to the ground, while short waves move eastward. Writing in the beach sand, Rossby derived their group velocity,
With cg > c the group velocity exceeds the phase velocity. The energy, far from moving slower than the waves, as with ocean waves, will in the atmosphere move ahead of them, faster than the mean flow (Fig. 4).
Having written his equations in the beach sand, Rossby suddenly discovered that the rising tide was about to wash them away. He rushed into the nearest restaurant and phoned Walter Munk, who hurried down to the beach. While Rossby explained his new theory, he barely managed to keep ahead of the rising water (W. Munk 1994, personal communication). In December 1945 (on Christmas Eve!), Rossby finalized to the Journal of Meteorology what would become the classical paper “On the Propagation of Frequencies and Energy in Certain Types of Oceanic and Atmospheric Waves” (Rossby 1945). Being “a very intuitive investigator” (Walter Munk, personal communication 1994) Rossby wanted to find out with what physical processes he was dealing. He was reluctant to substitute physical understanding with mathematical formalism. How did this group velocity manifest itself physically in the atmosphere? To discuss in terms of superposition of sine waves, as was often done, would be a “recourse to artificially induced interference patterns” (Rossby 1945, p. 188, 202). The nearest he could think about was downstream propagation of pressure falls and occasional upstream propagation of blocked patterns. It is now when Ernest Hovmöller enters the story.
THE BIRTH OF THE HOVMÖLLER DIAGRAM.
It was when Rossby at the SMHI lecture discussed how energy released in storms was transported with the group velocity—faster than the speed of the storm itself—that a bell rang within Hovmöller. A senior Danish forecaster, Leo Lysgaard, had made him aware that when there is an intense cyclogenesis west of Ireland, it is very probable that a strong high pressure system is created 1–2 days later over central Europe. This synoptic rule of thumb from the 1930s was documented by the Norwegian meteorologist Sigurd Evjen (Evjen 1936). Two of Rossby’s associates, Jerome Namias and Philip Clapp, also observed similar synoptic behaviors in upper-air patterns over North America. An intensification of a storm in the Gulf of Alaska was frequently followed by a downstream strengthening of a high pressure system over the western United States and was in a few days followed by a new low pressure system developing downstream over the eastern United States. This downstream development process profoundly affected the circulation over extensive parts of the hemisphere (Namias and Clapp 1944, p. 65). A follow-up study by Cressman (1948), using data from autumn 1945, confirmed the existence of such downstream developments.
To determine whether Rossby’s concept of group velocity was applicable, Hovmöller constructed a time–longitude diagram with the mean 500-hPa geopotential between 35° and 60°N depicted for every tenth longitude. He had already used a similar time–latitude diagram in a previous publication (Hovmöller 1947) about the weather in the Arctic waters. He had developed the idea about time–latitude graphs from the monthly summaries issued by the British Meteorological Office in the 1930s. His own time–longitude or trough–ridge diagram, as Hovmöller called it, clearly showed the large-scale upper-air wave pattern with stationary planetary waves and shorter transient waves. But it also showed something new: occasions with successive amplifications of the pressure systems moving rapidly with a speed of 25°–30° day−1 eastward, just as Rossby’s group velocity theory had predicted. Like Evjen (1936), Hovmöller never found such developments moving westward (upstream). Rossby became very enthusiastic about Hovmöller’s diagram and asked him to give a full account in his new journal Tellus (Hovmöller 1949). Rossby also incorporated its contents into an article of his own (Rossby 1949).
THE SCIENTIFIC IMPORTANCE OF THE HOVMÖLLER DIAGRAM.
Hovmöller’s diagram strongly supported some of the theoretical ideas in Rossby’s Chicago School on the interactions between individual cyclonic systems in particular (Rossby 1949; Yeh 1949; Gambo 1951). The “Bergen School” and other models, such as Sutcliffe’s development theory (Sutcliffe 1947) were local and unable to provide useful guidance in those situations. One of Rossby’s main actions at SMHI had been to increase the contacts with the meteorological world. The daily weather briefings soon took on an extra dimension when they were attended by the world’s leading scientists, such as Jule Charney, Herbert Riehl, Sverre Petterssen, Jerome Namias, and Reginald Sutcliffe. The new dynamical concepts were picked up by the operational forecasters, who integrated them into their synoptic analysis and applied them in useful ways in their routine work (Berson 1991). In particular Namias made a great impression on the Swedish forecasters at SMHI owing to his ability to link theory with practice. During a visit in spring 1949, Namias, one day while using Chicago School concepts during the map discussion, predicted that a strong cyclonic activity developing over the North Atlantic would lead to a quasi-permanent ridge of high pressure over northern Europe and make the upcoming Easter (14–17 April) sunny. The press got hold of the story and when, to everybody’s delight, Namias’s forecast was verified, Rossby’s barotropic approach became firmly rooted in the Swedish meteorological forecast culture for decades to come (Roads 1986, 16–17).
The new group velocity theory would, however, for a time have a profound impact on the emerging science of numerical weather prediction (NWP). In a 29 October 1948 letter to George W. Platzman, Rossby pointed out that group velocity and energy dispersion dealt with “the very heart” of the project of computerized forecasts (Platzman 1979; G. W. Platzman 1994, personal communication). It clarified the fact that the barotropic models were not just advecting wave patterns but were able to modulate these waves in a nontrivial way (Persson 2005). The notion of group velocity also helped to define sufficiently large computational areas. Because of the very limited computer power at the time, numerical forecasts had to be run on limited areas with constant boundary values. These artificially constant boundary values propagated into the heart of the computational area by the group velocity and distorted the forecasts (Phillips 1990, p. 4). Fortunately, Charney with some ambivalence (Phillips 2000, p.17) was able to apply group velocity arguments in a quantitative manner so that a reasonable decision could be made about the minimum size of the required forecast area. Unfortunately, some other NWP centers failed to realize this (for details see Persson 2005).
THE SYNOPTIC “GROUP VELOCITY THINKING.”
Despite some prominent American studies in the early 1950s on downstream development (Austin et al. 1953; Reed and Sanders 1953; Austin 1954), the interest in group velocity and downstream development seemed to wane in the scientific meteorological community, with van Loon (1965) as an exception. The interest stayed, however, alive among the forecasters. When I joined the SMHI In the late 1960s, Hovmöller diagrams were plotted on a daily basis for the 5-day forecast section together with the zonal 500-hPa zonal wind. They depicted the long planetary waves and the corresponding hemispheric wavenumber N (normally 2–6), which was inversely proportionally to the zonal flow U.
Based on Chicago School methods (Riehl 1951, 1952), this information provided guidance for purely manual 5-day forecasts. If the strength of the zonal 500-hPa wind showed a tendency to decrease (or increase), then the number of waves was supposed to increase (or decrease). Assuming that the so-called anchor troughs on the western side of the North Pacific and North Atlantic Oceans remained in their positions (forced from below by thermal SST contrasts and lee effects from upstream mountain ranges), it was possible to roughly infer the approximate positions of the other intermediate waves.
It often happened that the forecasters at the afternoon weather discussions started their overview of the synoptic situation far out in the North Pacific Ocean 5 days earlier. Instances of downstream developments, seen on the Hovmöller diagrams entering into the North American continent and the North Atlantic, were extrapolated into the European region and suggested possible changes of weather regimes. The resulting manual 5-day forecasts displayed predictive skill. From September 1965, SMHI confidently presented them twice a week on national television after the main news. This was some years before computer-based numerical 5-day barotropic forecasts were produced at SMHI. Even when the operational NWP became more skillful, Hovmöller diagrams remained in operation. The NWP solutions did not always adhere to the Bergen School synoptic rules. Because of downstream development, cyclones developed in the computer forecast could turn up in unexpected places. Hovmöller’s diagrams cautioned the forecasters to apply group velocity thinking and not to discard odd NWP solutions out of hand (Fig. 5). A study by some students (Eriksson et al. 1977) found that the Swedish numerical forecast system (baroclinic up to +48 h, then barotropic) slightly underforecasted the group velocity, 20°–25° instead of 25°–30° day−1 in the 30°–70° latitude band.
AN INTERNATIONAL COMEBACK FOR THE DIAGRAM.
In the late 1970s, three young British meteorological scientists at Reading University (United Kingdom) appealed to the theory of atmospheric energy dispersion when they tried to understand a numerical experiment on baroclinic instability, in particular the triggering of successive baroclinic waves (Hoskins et al. 1977). Contrary to Evjen (1936) and Hovmöller (1949), but in line with Rossby (1945) and Yeh (1949), Simmons and Hoskins (1979) found theoretical indications of upstream developments. They suggested these might have been related to the growth of secondary depressions and the formation of cyclone families on trailing cold fronts. Their papers could have stimulated further research, but soon after Simmons left for ECMWF and Hoskins headed toward new theoretical challenges.
The interest in Hovmöller diagrams to diagnose downstream development might have faded again had it not been for a series of papers in the late 1980s and early 1990s by Isodoro Orlanski and his student Edmund Chang, then at Geophysical Fluid Dynamics Laboratory (GFDL) at Princeton University. Chang and Orlanski (1993) depicted in Hovmöller diagrams the eddy vertical mean kinetic energy, and Chang (1993) also showed the meridional wind component at 300 hPa, which was first suggested by Carlin (1953).
Whereas the classical trough–ridge Hovmöller diagram (Fig. 2) highlights the geopotential centers, the Orlanski–Chang diagram, as it was sometimes called at ECMWF, highlights the geopotential gradients (Fig. 6).
The Orlanski–Chang diagram’s focus on wind speed (or kinetic energy) made the downstream development appear more clearly. It conveyed a stronger impression of the downstream domino effect as a matter of successive conversions between potential and kinetic energy. Kinetic energy, released from conversion of potential energy throughout the troposphere at the entrance region of the jet stream, is then rapidly transported downstream by the upper-tropospheric flow to the exit region of the jet stream, where it is then partly converted back to potential energy (Fig. 7). The upper-tropospheric flow as a conveyor belt for energy had some theoretical justification already in Rossby’s classic 1945 paper. Combining the abovementioned Eqs. (1a) and (1b), he found the simple equation [Eq. (128) in his notation]
implying that “the energy would be dispersed eastward at a speed in excess of U” (Rossby 1945, p. 198). Some years later, benefitting from the increased knowledge of upper-midlatitude tropospheric winds, it would have been possible to see these winds (typically 25–30 m s−1) were, just as Eq. (2) predicted, almost double the average zonal flow in the midtroposphere (typically 15 m s−1).
Hovmöller diagram techniques were also used to measure the typical speed of downstream development by Fraedrich and Lutz (1987) and later Lee and Held (1993). The impression about a relatively steady group velocity, varying at most between 25° and 30° day−1, was in contrast to the very varying nature of phase speeds, typically ranging between −5° and 20° day−1 (Fig. 8). It may therefore be conjectured that group velocity might be physically more fundamental than phase velocity. The same point, that phase velocity is of little significance compared to group velocity, has been made by Phillips (1990, p. 105), Landau and Lifshitz (1959, p. 263), and Lamb (1932, 383–84). Perhaps we are misled to believe the opposite just because the textbooks derive the group velocities from dispersive wave equations.
In the ECMWF Meteorological Operations (Met Ops) Room, Hovmöller diagrams and Orlanski–Chang diagrams were used for the daily operational monitoring of the medium-range forecast system, in particular to trace the origin of the bad forecasts—the forecast busts—insofar as they were due to poor initial conditions. Backtracking of errors causing forecasts busts using group velocity thinking, assuming a typical spread of influence of 25°–30° day−1, naturally focused the attention far upstream. The forecast errors occurring over Europe at day +3 typically had their origin over the westernmost Atlantic or eastern North America, whereas day +5 busts had their origin over western North America or the eastern North Pacific.1
The successful application of group velocity thinking motivated the head of the ECMWF Data Division, Phillippe Courtier, in October 1992 to suggest to do with mathematics what the meteorologists in the map room are doing with their eyes. The first publication (Rabier et al. 1993) was followed by many more that introduced the concept of “adjoint sensitivities.” These came to play an important role in experiments with “targeted observation” [Joly et al. 1997; Szunyogh et al. 2002; for an overview see Majumdar (2016)]. Yet another ECMWF application of Hovmöller diagrams at ECMWF was with the ensemble forecast system, in particular to present the forecast anomalies.
CURRENT USES OF HOVMÖLLER DIAGRAMS.
It would have been natural to end this article with a brief overview of how Hovmöller diagrams have come to good use in different applications during the last decades.2 The search engines on the American Meteorological Society and Wiley websites make it possible to count the number of papers published with a reference to Hovmöller diagrams (in any possible spelling). While the number of articles for a long time were about three papers per year, they have since the early 1990s (when the ECMWF displayed it in its map room) increased, more or less linearly, to more than 150 papers per year in 2015. Time has not allowed finding out whether the Hovmöller diagrams in all these papers have been time–longitude diagrams (not time–latitude ones). Nor has it been possible to find out how many of the genuine Hovmöller diagrams have been used for its original purpose: studying the downstream development process.
The Hovmöller time–longitude diagram might today be looked upon as just a convenient way to represent temporal evolutions of meteorological fields. But at its creation, the trough–ridge diagram played a crucial role in the development of early NWP. It convinced the main actors Jule Charney, John von Neuman, Carl Gustaf Rossby, and others that the theoretically derived group velocity (or signal velocity) were reflections of real physical–dynamical processes that had to be taken into account in the design of the systems. In his 1990 World Meteorological Organization (WMO) monograph on energy dispersion in atmospheric models, Norman A. Phillips (Phillips 1990, p. 4), one of the NWP veterans, reminded the reader that
if Charney and his collaborators had chosen too small an area in which to make their computations, the first modern attempt at numerical weather prediction would have been severely degraded by the spread of errors from outside the small forecast area… On the other hand, if a needlessly large area had been selected, the limited capacity of the electronic computer might have been exceeded.
When Ernest Hovmöller’s 90th birthday was celebrated at at the 23rd Nordic Meteorological Meeting in Copenhagen in 2002, he admitted he got the idea for his diagram by “a lucky hit.” In retrospect it can be said it was a lucky hit for the future development of meteorology.
The research for this article has passed three stages: the first stage in 1994/95, with valuable input from Brian Hoskins, Ernest Hovmöller, Karl-Einar Karlsson, Walter Munk, Norman Phillips, and Adrian Simmons; a second stage in 2002, as preparation for the ninetieth birthday celebration of Ernest Hovmöller in Copenhagen with a more in-depth interview; and finally in 2016 for this article, with additional useful information from Edmund Chang, Florence Rabier, Gösta Sjölander, and Zoltan Toth. A special thanks to Tim Coleman for the very useful comments during the final stages of preparation of the paper.