The present chapter begins with the study of ice particle collision with a solid wall (Section 14.1) and the consequences of such a collision, e.g. particle attrition and splitting (Section 14.2). The fracture of the target and projectile during normal penetration is described in Sections 14.3 to 14.5 using a model of chaotic disintegration modifying the theory of chaotic disintegration of liquids. The radius of the locally smallest fragment is calculated equating its kinetic energy of deformation with its surface energy of fracture. The probability of lacunae opening in the target and projectile materials increases near the target/projectile interface. The percolation threshold for this probability determines the boundary of the fractured zone. When this fractured zone reaches the rear surface of the target, the fragments can leave it. Mass distribution of the fragments was calculated with the help of percolation theory. Then, the shape of the debris cloud and the direction, velocity and range of its propagation are calculated to estimate vulnerability behind the perforated target (Section 14.6). The effect of plastic dissipation on debris sizes is estimated in Section 14.7.

Ice Particle Collision with a Dry Solid Wall

The reason for interest to better understand ice particle impact is mainly in the attempt to model and predict potential damage which such impact can cause on solid structures. It is important for ship building and the design of aircraft, arctic and space research. Moreover, ice crystal impact in hot environments, e.g. in aircraft engines or on heated measurement instruments can lead to ice accretion. Melting and subsequent shedding of the accumulated ice layer can result in even greater damage, e.g. impact onto aircraft compressor stages.

Experimental investigation of impact in laboratories using artificial, simulated ice particles is the main source of the knowledge about mechanisms of particle deformation and breakup (Hauk et al. 2015). The impact is observed using high-speed video systems. Many studies are focused on characterization of the kinematics of the post-impact fragments (Vargas et al. 2014, Emery et al. 2004, Vidaurre and Hallett 2009, Guégan et al. 2011). In particular, the restitution coefficient of the ice particles is measured (Hatzes et al. 1988, Higa et al. 1998). Images of a high-speed hail ice impact showing details of its deformation and fragmentation, captured using a high-speed video system, can be found in Tippmann et al. (2013).

The first two sections in the present chapter are devoted to penetration of shaped-charge (Munroe) jets. Shaped-charge jets are characterized by such tremendous stresses that metal flow can be treated in the framework of potential flow hydrodynamics. Section 13.1 is devoted to the elementary theory of penetration depth of a shapedcharge jet perforating an armor, while Section 13.2 describes a detailed structure of the corresponding metal flow and predicts the crater shape using the hodograph method of complex analysis in the planar case. An estimate for an axisymmetric case is also given in this section. Normal penetration of eroding projectiles into elastic–plastic targets is covered in Section 13.3. The limit of high-speed penetration where pressure and the inertial effects are dominant is considered in Section 13.4, and the quasi-steady regime of penetration of an eroding projectile is treated in Section 13.5. Section 13.6 is devoted to description of normal and oblique penetration of rigid projectiles and comparison with numerous experimental data. Explosion welding discussed in Section 13.7 is also characterized by tremendous stresses, while the resulting metal flow is not only “inviscid” but even compressible. Still, the problem can be reduced to the form appropriate for complex analysis, which reveals some important details of the interface formation between two materials welded by this method. The experimental evidence shows some additional fascinating details which deserve a further analysis.

Shaped-charge Jet Penetration Depth

In 1888, Munroe discovered the phenomenon of so-called shaped-charge (or Munroe) jets. Later on, he described it in the following words (Munroe 1900):

“Among the experiments made … was one upon a safe twenty-nine inches cube, with walls four inches and three quarters thick, made up of plates of iron and steel … When a hollow charge of dynamite nine pounds and a half in weight and untamped was detonated on it, a hole three inches in diameter was blown clear through the wall … The hollow cartridge was made by tying the sticks of dynamite around a tin can, the open mouth of the latter being placed downward.”

EQUATORIAL OCEANOGRAPHY DECEIVES US, hiding fascinating, non-intuitive dynamics beneath the languorous tropical air. The mid-latitudes give us the great gyres with their intense western boundary currents and mesoscale eddies, and by comparison the equatorial currents may seem, on the surface, featureless and vapid. Yet the equatorial regions are home to the resolute equatorial undercurrents that tunnel across the basins, opposite in bearing to the winds that drive them. And the equatorial ocean and atmosphere — in a collaboration that is more tango than waltz — give rise to the marvellous phenomenon that is El Niño, the most dramatic example of climate variability on human timescales that this planet has to offer. Such phenomena are the subjects of this chapter.

The defining feature of equatorial dynamics is that the Coriolis parameter becomes small, at least by comparison with the mid-latitudes, and balanced and unbalanced dynamics become intertwined, as we encountered in Chapter 8. Yet if we move more than a few degrees away from the equator the Rossby number again becomes quite small, suggesting that familiar ways of investigating the dynamics — Sverdrup balance for example — might yet play a role. Let's first see what we are trying to understand and if the observations can give us some intuition.

OBSERVATIONAL PRELIMINARIES

In mid-latitudes the gyres are very robust features, existing in all the basins, and may be understood as the direct response to the curl of the wind stress. In the equatorial regions the currents also display some robust and distinctive features, illustrated in Fig. 22.1 and the top panel of Fig. 22.2, but their relation to the winds is less obvious. The main features are as follows:

1. 1. A shallow westward flowing surface current, typically confined to the upper 50 m or less, strongest within a few degrees of the equator, although not always symmetric about the equator. Its speed is typically a few tens of centimetres per second.

2. 2. A strong coherent eastward undercurrent extending to about 200 m depth, confined to within a few degrees of the equator. Its speed is up to a metre per second or a little more, and it is this current that dominates the vertically integrated transport at the equator. Beneath the undercurrent the flow is relatively weak.

UNDERSTANDING THE CIRCULATION OF THE OCEAN involves a combination of observations, comprehensive numerical modelling, and more conceptual modelling or theory. All are essential, but in this chapter and the ones following our emphasis is on the last of the triad. Its (continuing) role is not to explain every feature of the observed ocean circulation, nor to necessarily describe details best left to numerical simulations. Rather, it is to provide a conceptual and theoretical framework for understanding the circulation of the ocean, for interpreting observations and suggesting how new observations may best be made, and to aid the development and interpretation of numerical models.

The aspect of the ocean that most affects the climate is the sea-surface temperature (sst), as illustrated in Fig. 19.1, and aside from the expected latitudinal variation there is significant zonal variation too — the western tropical Pacific is particularly warm, and the western Atlantic is warmer than the corresponding latitude in the east. These variations owe their existence to ocean currents, and the main ones are sketched — in a highly schematic and non-quantitative fashion — in Fig. 19.2. Over most of the ocean, the vertically averaged currents have a similar sense to the surface currents, one exception being at the equator where the surface currents are mainly westward but the vertical integral is dominated by the eastward undercurrent. Two dichotomous aspects of this picture stand out: (i) the complexity of the currents as they interact with topography and the geography of the continents; (ii) the simplicity and commonality of the large-scale structures in the major ocean basins, and in particular the ubiquity of subtropical and subpolar gyres. Indeed these gyres, sweeping across the great oceans carrying vast quantities of water and heat, are perhaps the single most conspicuous feature of the circulation. The subtropical gyres are anticyclonic, extending polewards to about 45°, and the subpolar gyres are cyclonic and polewards of this, primarily in the Northern Hemisphere. The existence of the great gyres, and that they are strongest in the west, has been known for centuries; this western intensification leads to such well-known currents as the Gulf Stream in the Atlantic (charted by Benjamin Franklin), the Kuroshio in the Pacific, and the Brazil Current in the South Atlantic.

WATER IS AN ORDINARY SUBSTANCE WITH EXTRAORDINARY EFFECTS. The most obvious is that oceans themselves are made of water, and if our planet were dry this book would perforce be much shorter (if only). Leaving aside the dynamical effects of the oceans, water covers over two-thirds of Earth's surface and because it is warm in some places and cold in others, and because the atmosphere is in motion, water evaporates into the atmosphere in one place and condenses from it elsewhere. The condensation leads to rain, one of the most talked-about aspects of weather and climate. Water also freezes to form ice, so that at any given time water exists on Earth in all three phases. Radiatively, water vapour is a greenhouse gas, meaning that it absorbs infrared radiation that might otherwise be lost to space and so maintains the surface of the planet at a temperature over 20 K higher than an equivalent dry planet. Dynamically, the condensation of water vapour in the atmosphere releases energy, warming the air and tending to make it more unstable than otherwise and leading to convection. Further, the net transport of water vapour from low to high latitudes is effectively a meridional transport of energy.

In this chapter we focus on a small number of these issues, mainly on the kinematics and dynamics of water vapour itself and on some aspects of the dynamics of the tropical atmosphere, where the effects of water vapour are most manifest. The tropics would certainly differ from the mid-latitudes even if the atmosphere were dry — its Coriolis parameter is small among other things — so our attention there is by no means confined to the effects of water vapour. Nevertheless, tropical convection and the attendant ‘radiative-convective equilibrium’ are greatly influenced by the presence of water. We begin with a discussion of the thermodynamic properties of water vapour itself. We then move on to an essentially kinematic description of the factors determining the large scale distribution of relative humidity, before finally looking at convection and at tropical dynamics more generally.

A MOIST IDEAL GAS

Water is the compound of hydrogen and oxygen with the chemical formula H2O, although in informal conversation water is often understood to mean only the liquid form of the compound. Water vapour is a gas made up of molecules of H2O, and ice is the solid form of water.