Introduction

In the previous chapter we introduced the idea of small deformation, which allowed us to neglect the distinction between the Lagrangian and the Eulerian description. Now we will apply the small-deformation hypothesis to the equation of motion (3.5.3). The resulting equation will include spatial derivatives of the stress tensor, the acceleration of the displacement, and body forces. The displacement, in turn, is related to the strain tensor via (2.4.1). Therefore, we have two systems of equations, one for stress and displacement and one for strain and displacement. Within the approximations that have been introduced, these equations are valid for any continuous medium. To apply them to a specific type of medium (e.g., solid, viscous fluid) it is necessary to establish a general relation (known as a constitutive equation) between stress and strain.

In the case of solids, when a body is subjected to external forces it becomes deformed (strained), and internal stresses are generated within the body. The relation between stress and strain depends on the nature of the deformation and other external factors, such as the temperature. If the deformation is such that the deformed body returns to its original state after the force that caused the deformation is removed, then the deformation is said to be elastic. If this is not the case, i.e., if part of the deformation remains, the deformation is known as plastic.

Introduction

The theory developed so far is not completely realistic because it does not account for the observed fact that the elastic energy always undergoes an irreversible conversion to other forms of energy. If this were not the case, a body excited elastically would oscillate for ever. The Earth, in particular, would still be oscillating from the effect of past earthquakes (Knopoff, 1964). The process by which elastic energy is lost is known as anelastic attenuation, and its study is important for several reasons. For example, because attenuation affects wave amplitudes and shapes, it is necessary to account for their variations when computing synthetic seismograms for comparison with observations. Properly accounting for the reduction in wave amplitude was particularly important during the cold-war period because of the use of seismic methods to estimate the yield of nuclear explosions in the context of nuclear test-ban treaties. In addition, because attenuation depends on temperature and the presence of fluids, among other factors, its study has the potential for shedding light on the internal constitution of the Earth. The study of attenuation may also help us understand the Earth's rheology, although the relation between the two is not clear. For a discussion of these and related matters see Der (1998), Karato (1998), Minster (1980), and Romanowicz and Durek (2000).

From a phenomenological point of view, the effect of attenuation is a relative loss of the high-frequency components of a propagating wave.

Aspects common to both polar regions

Introduction

Polar lows can have a severe impact on maritime operations and can cause considerable disruption when the more severe systems make landfall. In areas such as the North Sea, there are many gas and oil platforms and it is necessary to have good forecasts of the arrival of severe mesoscale lows to minimize the impact on operations. Within the Antarctic, most of the mesoscale lows are not as vigorous as their counterparts in the north. Nevertheless, they can still cause severe problems during the summer relief operations at the research stations and affect work in the deep field.

Polar low forecasting is an integral part of the general forecasting problem and the results are dependent on the success of the overall forecast. Polar lows often result in a rapid deterioration of the weather at a specific location, and accurate time indications in the forecasts are important. In this chapter we will examine the means by which forecasters attempt to predict the formation and development of mesocyclones and polar lows. The output from numerical weather prediction (NWP) analysis and forecast systems can be used to try and infer where and when mesocyclone developments may take place in a particular region. Satellite imagery and other satellite data are indispensable in identifying and predicting the movement of existing mesoscale vortices a few hours ahead.

It will have become clear from the preceding chapters that major advances have been made over the last few decades in our understanding of the distribution, occurrence, formation and development mechanisms of polar lows and other high latitude mesoscale vortices. In this final chapter we summarize our present understanding of this family of weather systems and consider the requirements for future research.

The spatial distribution of polar lows and other high latitude mesoscale vortices

Polar lows were first investigated over the Nordic Seas and when systematic research started into such systems in the 1960s the case studies were concerned with lows occurring in this region. Subsequently, such investigations were gradually extended to new areas in the Northern Hemisphere, such as the Labrador Sea, the Gulf of Alaska and the Sea of Japan, where similar active mesoscale vortices were identified. Since the 1970s isolated cases have also been reported of polar lows over the marginal seas around the Arctic Ocean, including the Beaufort Sea, the Chukchi Sea and the Kara Sea. However, it is now unlikely that significant numbers of active systems will be found in any new areas in the Northern Hemisphere.

In the Southern Hemisphere, polar lows have only been thoroughly investigated in a limited number of regions. The lows found around New Zealand seem to have many of the characteristics of the systems occurring in the Nordic Seas, but much more work needs to be done on the polar lows in this area.

Polar lows and other mesoscale lows in the polar regions

In this volume we are concerned with the whole range of mesoscale lows with a horizontal length scale of less than c. 1000 km that occur in the Arctic and Antarctic poleward of the main polar front or other major frontal zones. However, much of the interest will be focused on the more intense systems, the so-called polar lows. The term mesocyclone covers a very wide range of weather systems from insignificant, minor vortices with only a weak cloud signature and no surface circulation, to the very active maritime disturbances known as polar lows, which in extreme cases may have winds of hurricane force and bring heavy snowfall to some areas. Clearly it is very important to be able to forecast these more active systems since they can pose a serious threat to marine operations and coastal communities when they make landfall.

Although it has been known for many years in high latitude coastal communities that violent small storms could arrive with little warning, it was only with the general availability of imagery from the polar orbiting weather satellites in the 1960s that it was realized that these phenomena were quite common. The imagery indicated that the storms developed over the high latitude ocean areas (generally during the winter months) and tended to decline rapidly once they made landfall.

Introduction

During the 1970s, research into the theoretical understanding of high latitude mesocyclones was focused on the basic mechanisms of development of the more intense systems, known as polar lows. The aim was to explain the striking differences between polar lows and other extra-tropical cyclones, namely the small size and rapid growth rates of polar lows, and their favoured formation within cold air masses over the oceans in winter. It will become apparent by the end of this chapter that these fundamental questions have not been completely answered. However, considerable progress has been made, and new areas of research have been opened up regarding the life-cycle of polar lows, and their inter-action with the broadscale atmospheric flow.

The construction of mathematical and theoretical models of mesocyclones is not simple, because there are many types of vortices occurring in the high latitude areas. They vary widely in horizontal and vertical extent, in intensity and in structure. A mesocyclone may be a powerful system, extending through the depth of the troposphere, with intense deep convection and hurricane-force winds, or a weak swirl in the boundary-layer cloud, clearly visible on satellite imagery but with little significant weather at the Earth's surface. The environment in which the vortex forms may differ widely being, for example, a low-level frontal zone, or a flaccid low-pressure region at the centre of a decaying synoptic cyclone.

The Arctic

Introduction

For several decades observational investigations in the form of case studies have supplied an important part of the attempts to understand the structure and development of mesoscale vortices. Apart from obtaining a description of the individual cases, an underlying purpose has been, through a synthesis of the different cases, to gain sufficient knowledge to describe the basic properties of these systems, including their structure and dynamics. Present-day high resolution numerical models have proved to be very effective for simulating the structure and development of mesoscale systems, such as polar lows in data sparse regions, and case studies in the form of model simulations of polar low developments have yielded much important information about these systems. The results from these studies will be discussed separately in Chapter 5, but also, when relevant and where model studies have been coupled with observational investigations, in this chapter.

A very significant part of the polar low research over the last 30 years has been dedicated to the Nordic Seas (defined as the North Atlantic east of Greenland and north of 60° N, plus the North Sea, the Norwegian Sea, the Greenland and Barents Seas), which is a primary genesis region for polar lows. The following discussion will start therefore by presenting the results from research carried out in this region. This discussion will be followed by an overview of parallel work carried out in other parts of the Northern Hemisphere, including important results obtained by Japanese researchers.

The Arctic

Introduction

A large number of the most significant mesoscale vortices/polar lows form in northerly flows close to the Arctic coast or along the ice edges bordering the coast. For this reason, knowledge of the general weather and climatic conditions in the Arctic region is of major importance for an understanding of the formation of mesoscale cyclones, including polar lows.

As explained in Chapter 1, polar lows are a subclass of especially intense, maritime cyclones among the more general mesoscale cyclones. In Scandinavia and elsewhere in northwestern Europe, the main interest for many years has been focused upon the more intense systems, i.e. the polar lows. The term has been widely accepted throughout the meteorological community in the region, even for systems which do not, in a strict sense, fulfil the requirements of a wind speed around or above gale force. For this reason in this section we will generally use the term ‘polar low’ instead of the more general ‘mesoscale cyclone’ or ‘mesocyclone’.

The Arctic region is dominated by the huge, generally sea ice-covered Arctic Ocean. It is approximately as large as the Antarctic continent, but apart from this, there are striking differences between the two regions (see Section 2.2). The Arctic Ocean surrounding the North Pole is bordered by Scandinavia, Siberia, Alaska, Canada and Greenland. It consists of a large basin, the Arctic Ocean, plus a number of marginal seas along the continental shelves.

The Arctic

Introduction

Numerical models of the atmosphere, which predict future conditions from an analysis of an initial state, have proved to be a tool of growing importance in the study and forecasting of polar lows. To some degree, the life cycles of some polar lows are now simulated operationally by numerical weather prediction (NWP) centres. Some special cases of polar lows have been simulated more extensively in an a posteriori, non-operational mode with models suited for this purpose. Such simulations have provided a new form of data for the study of the formation and evolution of these vortices.

The history of the development of NWP, along with the growth of computing capacity in super-computers, is well known. For a long time the resolution of the numerical models was too coarse to describe polar lows. A breakthrough for the simulation of polar lows came in a polar low project organized by the Norwegian Meteorological Institute (DNMI) in the first half of the 1980s (Lystad, 1986; Rasmussen and Lystad, 1987). As part of this project, a mesoscale NWP system was established (Grønås et al., 1987b; Grønås and Hellevik, 1982; Nordeng, 1986), which gave the first realistic numerical simulations of polar lows (Grønås et al., 1987a; Nordeng, 1987). As operational NWP systems were further developed, reliable guidance for the prediction of polar lows was eventually advanced at several meteorological centres.

Since the first detailed investigations of polar lows and other high latitude, mesoscale weather systems were carried out in the late 1960s there have been major advances in our knowledge regarding the nature of such systems and the mechanisms behind their formation and development. High resolution satellite imagery has shown how frequently such lows occur in both polar regions and has illustrated the very wide range of cloud signatures that these systems possess. Great strides have also been made in representing these weather systems in numerical models. With their small horizontal scale, it proved difficult to represent the lows in the early modelling experiments, but the new high resolution models with good parameterizations of physical processes have been able to replicate a number of important cases, despite the lack of data for use in the analysis process.

Although case studies of mesoscale lows have been undertaken for many years, recent research has been able to draw on many new forms of data, especially from instruments on the polar orbiting satellites. Scatterometers have provided fields of wind vectors over the ice-free ocean, passive microwave radiometers have allowed the investigation of the precipitation associated with the lows, and new processing schemes for satellite sounder data have given information on their three-dimensional thermal structure. In addition, aircraft flights through polar mesoscale lows have provided high resolution, three-dimensional data sets on the thermal and momentum fields.

This chapter is concerned with the main topic of the monograph, namely, the solution of the GRP for quasi-1-D, inviscid, compressible, nonisentropic, time-dependent flow. In Section 5.1 we formulate the problem and study its solution in the Lagrangian and Eulerian frames. In particular, we state and prove the main ingredient in the GRP method, Theorem 5.7. A weaker form of this theorem leads to the “acoustic approximation” (Proposition 5.9). Summary 5.24 gives a step-by-step description of the GRP analysis. In Section 5.2 we present the GRP methodology for the construction of second-order, high-resolution finite-difference (or finite-volume) schemes. Starting out from the (first-order) Godunov scheme, we present the basic (E1) GRP scheme. It is based on the acoustic approximation and constitutes the simplest second-order extension of Godunov's scheme. This is followed by a presentation of the full array of GRP schemes (as well as MUSCL). Generally speaking, the presentation in this chapter follows closely the GRP papers [7] and [10].

The GRP for Quasi-1-D, Compressible, Inviscid Flow

In Section 4.2 we studied the Euler equations (4.45) governing the quasi-1-D flow in a duct of variable cross section. We emphasized in particular the role of the Riemann problem (“shock tube problem”), namely, the IVP subject to initial data (4.100). As we shall see in this chapter, the solution to the Riemann problem is a basic ingredient in the numerical resolution of the flow.