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.

In Definition 2.15 we gave the most practical version of the entropy condition. It limits admissible shocks to those obtained by the intersection of “forward-moving” characteristics. These are therefore discontinuities that “cannot be avoided” or replaced by a rarefaction wave. In this Appendix we give some further insight into this concept of an “entropy satisfying” weak solution to (2.1), (2.2).

Our starting point is the physical notion of a “vanishing viscosity solution.” In general terms, an equation leading to discontinuous solutions [such as (2.1)] is supplemented by “dissipative terms” (also referred to as “viscous terms”). In analogy to the physical situation, such terms have a “smoothing effect” on solutions with large gradients, thus replacing discontinuities by “transition zones” where the solution varies smoothly, albeit rapidly. As the viscous effects are diminished, those transition zones shrink to surfaces of zero width, across which the solution has a sharp jump. Mathematically speaking, the additional viscous terms are often represented by second-order derivatives with a small (“vanishing”) coefficient.

To illustrate the situation, we consider the “moving step” problem for Burgers' equation (Example 2.12).

The phenomenological theory discussed in the previous chapter did not permit the parameterization of the energy dissipation. In this chapter spectral turbulence theory will be presented to the extent that we appreciate the connections among the turbulent exchange coefficient, the energy dissipation, and the turbulent kinetic energy. In the spectral representation we think of the longer waves as the averaged quantities and the short waves as the turbulent fluctuations. Since the system of atmospheric prediction equations is very complicated we will be compelled to apply some simplifications.

Fourier representation of the continuity equation and the equation of motion

Before we begin with the actual transformation it may be useful to briefly review some basic concepts. For this reason let us consider the function a(x) which has been defined on the interval L only. In order to represent the function by a Fourier series, we extend it by assuming spatial periodicity. Using Cartesian coordinates we obtain a plot as exemplified in Figure 12.1. The period L is taken to be large enough that averaged quantities within L may vary, i.e. the averaging interval Δx ≪≪ L.

Certain conditions must be imposed on a(x) in order to make the expansion valid. The function a(x) must be a bounded periodic function that in any one period has at most a finite number of local maxima and minima and a finite number of points of discontinuity.

Introduction

The vertical structure of the atmospheric boundary layer is depicted in Figure 13.1. The lowest atmospheric layer is known as the laminar sublayer and has a thickness of only a few millimeters. It is difficult to verify the existence of this layer because of its small vertical extent. Within the laminar sublayer all physical processes such as the transport of momentum and heat are regulated by molecular motion. In most boundary layer models the existence of this layer is not explicitly treated. It stands to reason that there also exists some type of a transitional layer between the laminar sublayer and the so-called Prandtl layer where turbulence is fully developed.

The lower boundary of the Prandtl or surface layer is the roughness height z0 where the mean wind is assumed to vanish. The vertical extent of the Prandtl layer is regulated by the thermal stratification of the air and may vary from about 20 to 100 m. In this layer all turbulent fluxes are approximately constant with height. The influence of the Coriolis force may be ignored this close to the earth's surface, so the turning of the wind within the Prandtl layer may be ignored. The wind speed, however, increases very strongly in this layer, reaching a value of more than half the wind speed at the top of the boundary layer.

This chapter addresses one of the most central issues of computational fluid dynamics, namely, the simulation of flows under complex geometric settings. The diversity of these issues is briefly outlined in Section 8.1, which points out the role played by the present extensions: the (1-D) “singularity tracking” and the (2-D) “moving boundary tracking” (MBT) schemes. Section 8.2 deals with the first extension, and Section 8.3 is devoted to an outline of the second one. In the former we present the scheme methodology and refer to GRP papers for examples. In the latter, the basic principles of the method are presented, and we refer to [39] for more algorithmic details. Finally, an illustrative example of the MBT method shows how an oval disk is “kicked-off” by a shock wave.

Grids That Move in Time

In Part I of this monograph we dealt with finite-difference approximations to the quasi-1-D hydrodynamic conservation laws, where the underlying grid was fixed and equally spaced in the majority of cases. In our two-dimensional numerical extension (Section 7.3) we restricted the treatment to a Cartesian (rectangular) grid. Naturally, finite-difference approximations assume their simplest form on such grids, and the motivation for seeking geometric extensions comes primarily from physical applications.