Ocean models are based on the conservation of vector momentum, and scalar tracers, which give time evolution (prognostic) equations for the velocity field, the active temperature and salinity tracers, and all passive tracers of interest, such as oxygen, carbon dioxide, and nutrients. In contrast, ocean density, ρ, is diagnostic and not necessarily conserved. It is computed from the active tracers using an equation of state (EoS). Griffies (2002) and works cited therein discuss the approximations leading to the primitive equations that are typically solved by climate models, and common ways that the global ocean is discretized. Various methods of integrating the equations are also presented, along with their advantages and disadvantages. Asummary, including the equations themselves, is provided in Treguier et al. (Chapter 7, this volume). The existence of such excellent references means that this chapter can focus on specific illustrative examples of the workings of buoyancy in particular ocean models, without being comprehensive, or repeating the background.

The great challenge of climate modeling is the roughly 10-decade-wide range of potentially important interacting scales; from the more than 107 m global scale to the less than 10-2 m viscous scale. Present coupled climate calculations are reaching down to the 104 m horizontal and 10 m vertical scales in the ocean, but still many subgrid scale (SGS) processes and interactions rely on parameterizations. In principle, model fidelity should benefit from increased resolution. Indeed, improvements can be dramatic.

Introduction

The Atmosphere

The atmosphere, like the ocean, is a stratified fluid highly influenced by the rotation of the Earth.

But, unlike the ocean, the atmosphere is a mixture of gases known as air.

The composition of the gas layer around the Earth has evolved very slowly since the time of its formation. Thanks to the appearance of life about 3.5 billion years ago, the main constituents of the air are now nitrogen (N2, about 78%) and oxygen (O2, about 21%). Other minor constituents are argon (1%), ozone, carbon dioxide, and water vapor.

The air near the surface is about 1,000 times lighter than the water in the ocean. It is also much more compressible. The mean state of the atmosphere is stably stratified and is in hydrostatic equilibrium (Figure 8.1). The first 10–15 km of the atmosphere, known as the troposphere, contain nearly 90% of the atmospheric mass. This is the layer where the weather occurs. The bottom of the troposphere, the boundary layer (BL), is directly influenced by the surface (land or ocean). Its mean depth is about 1 km, but depth can reduce to a few tens of meters on a cold winter day and expand to several kilometers on a warm, turbulent, summer day. The layer above the troposphere, called the stratosphere, is stably stratified. The stratosphere is the layer where the ozone chemistry protects the air and the surface below from incoming ultraviolet.

Introduction

Gravity currents occur when there are horizontal variations in density in a fluid under the action of a gravitational field. Asimple example that can be readily experienced is the gravity current that flows into a warm house through a doorway when it is opened on a cold, windless day. The larger density of the cold air produces a higher pressure on the outside of the doorway than on the inside, and this pressure difference drives the cold air in at the bottom and the warm air out at the top of the doorway. In addition to horizontal density variations, there must also be some feature to stop the fluid from either rising or falling indefinitely and to constrain the flow to be primarily horizontal. In many cases this is a solid boundary, such as the ground. In other situations it may be another feature of the density variations within the fluid, such as a density interface or vertical density stratification.

Gravity currents occur in gases when there are temperature differences, such as in the sea breeze, the flow of cool moist air from the sea to the land. On a warm day, the sun heats the land more than the sea, and, consequently, the air at low altitudes over the land is warmer than that over the sea. The sea breeze is a significant feature of coastal meteorology in many parts of the world.

General Considerations and a Laboratory Example

Introduction

Both the atmosphere and ocean are rotating and stratified, and for large-scale motions (an attribute that needs careful definition), both are important in shaping the dynamics. Both the mean circulation, whose scales usually reflect the forcing, and the inevitable eddy fields that result from the instability of those motions are determined by the effects of rotation and stratification. In this chapter, I am going to examine some particular aspects of the buoyancy-driven motion of a rotating stratified fluid with an eye to oceanic phenomena, in particular the important question of the relationship between the vertical motion and the surface buoyancy forcing. This question is of particular importance in the discussion of the oceans' role in climate since a key issue in delineating that role is how and where the sinking of surface-cooled water takes place. It should not be surprising that the presence of rotation and, in particular, the variation of the local normal component of that rotation (the beta-effect) renders the association of forcing and sinking sometimes nonintuitive. In sections 2.2 and 2.3, we will examine the nature of the circulation in simple models with an eye to uncovering in easily understandable circumstances the underlying physics that determines the structure of the vertical motion. Of course, the nature of the horizontal motion is coupled to the vertical motion and will be discussed as well.

Introduction

Particle-laden, gravity-driven flows occur in a large variety of natural and industrial situations. Typical examples include turbidity currents, volcanic eruptions, and sand-storms (see Simpson 1997 for a review). On mountain slopes, debris flows and snow avalanches provide particular instances of vigorous dense flows, which have special features that make them different from usual gravity currents. Those special features include the following:

• They belong to the class of non-Boussinesq flows since the density difference between the ambient fluid and the flow is usually very large, whereas most gravity currents are generated by a density difference of a few percent.

• Whereas many gravity currents are driven by pressure gradient and buoyancy forces, the dynamics of flows on slope are controlled by the balance between the gravitational acceleration and dissipation forces. Understanding the rheological behavior of particle suspensions is often of paramount importance when studying gravity flows on steep slope.

This chapter reviews some of the essential features of snow avalanches and debris flows. Since these flows are a major threat to human activities in mountain areas, they have been studied since the late 19th century. In spite of the huge amount of work done in collecting field data and developing flow-dynamics models, there remain great challenges in understanding the dynamics of flows on steep slope and, ultimately, in predicting their occurrence and behavior. Indeed, these flows involve a number of complications such as abrupt surge fronts, varying free and basal surfaces, and flow structure that changes with position and time.

Introduction

Volcanic systems are controlled by a wide range of fluid mechanical processes, including the subsurface migration and ensuing accumulation of molten rock in crustal reservoirs, know as magma chambers, and the subsequent explosive eruption of ash and transport high into the atmosphere (Sparks et al. 1997). The philosophy of this chapter is to develop a series of simplified physical models of the flow processes in order to gain insights about the dominant controls on the processes; many of the models have been developed based on geological field evidence and have been tested with laboratory experiments. The aim is to build understanding rather than to simulate the processes, which are extremely complex and often for which there is insufficient data for a complete characterization of the physical and chemical state of the system. We provide a brief introduction to the overall range of processes which occur, and then immerse ourselves in some of the fascinating buoyancy-driven flow processes. For a geological introduction to some of the processes and context, textbooks such as McBirney (1985) include a comprehensive description of many of the geological observations and processes.

In magma reservoirs, which may lie 5–10 km below the surface, magma is exposed to the relatively cold surrounding crust, and this may lead to cooling and crystallization of the magma, as well as melting of the surrounding crustal rock, often called country rock. In addition, new magma may be supplied to the magma reservoir leading to pressurization of the system.

Introduction

Dynamics of Water Mass Formation and Spreading

Small-scale buoyancy-driven flows, such as the overflows from marginal seas, are the main process by which the distinct water masses of the deep ocean are formed. For example, the flow of Antarctic Bottom Water (AABW) from the continental shelf down to the bottom of the Weddell Sea influences water mass properties all the way to the North Atlantic Ocean. The large range of spatial scales and mechanisms involved in the formation and spreading of these water masses poses a formidable challenge to numerical models. Legg (Chapter 5, this volume) reviews the main dense overflows of the world ocean. The width of an overflow is set either by the width of the strait or channel through which it flows (in the case of the Red Sea overflow, for example) or by the Rossby radius of deformation, which is the main dynamic scale for stratified rotating fluids. For an overflow of thickness h, with density anomaly δρ relative to the density ρ of the surrounding fluid, the reduced gravity g′ is defined as gδρ/ρ (g being the acceleration of gravity) and the Rossby radius Lr is defined as Lr = (g′h)1/2 /f with f being the Coriolis parameter. Lr decreases with latitude and its magnitude is only a few kilometers in the Nordic Seas. The dynamics of the plumes of dense water and the amount of entrainment that takes place as they descend along topographic slopes set the properties of the newly formed water masses (e.g., Chapter 5 by Legg).

Introduction

Marginal seas subject to buoyancy loss, because of their semi-enclosed geometry, are source regions for the formation of dense intermediate and bottom waters. These convective water masses generally have distinct properties relative to the open ocean and can be traced far from their formation basins. They also can transport significant amounts of heat, salt, and other tracers throughout the world ocean. The vertical circulation and meridional heat and freshwater transports are fundamental components of the oceanic circulation, and play important roles in the global climate system. Understanding how this circulation depends on the environmental parameters of the system is important if one is to better model and predict the climate system and its sensitivity to changing atmospheric conditions, such as increasing anthropogenic carbon dioxide.

The focus of this review is on the circulation and exchange resulting from surface buoyancy forcing in marginal seas. General characterisitics of the exchange between the marginal sea and the open ocean are described from eddy-resolving numerical models in idealized configurations, and the physics governing this exchange are elucidated through a combination of numerical models and simplified analytic models. Although the problems are couched in terms of marginal sea–open ocean exchange, many of the processes that emerge from this analysis are relevant to more general buoyancy-forced flows. Particular attention is paid to the dynamics involved with net vertical motions forced by surface cooling.