Utrecht summer school Physics of the climate system August 15-26, 2016 Rossby and Kelvin waves Institute for Marine and Atmospheric research Utrecht Utrecht University, Utrecht, The Netherlands Anna von der Heydt Phone: +31-30-2535963 fax: +31-30-2543163 e-mail: [email protected] 2 Contents 1 Introduction 2 2 Overview of wave types 2 3 Analytical preparations 3 3.1 The linearized shallow water equations . 3 3.2 The reduced gravity model . 4 3.3 Streamfunctions . 6 4 The Rossby wave 7 4.1 Structure and propagation characteristics . 9 4.2 The propagation mechanism of Rossby waves . 12 5 The Kelvin wave 13 1 1 Introduction In this lecture we will study waves in geophysical fluids. Waves are of great im- portance in atmospheric and oceanic flows, they transport information through the earth system in the fasted possible way. First, an overview of the various wave types is given. Second, the governing equations of so-called Rossby and Kelvin waves are derived. To this end, the shallow-water equations are linearized and analytical solutions are found. In their linearized form, the equations still represent the main behavior of these wave types. We restrict ourselves to Rossby and Kelvin waves here as the large-scale atmospheric and oceanic disturbances mostly generate these types of waves. Gravity waves and intertial gravity waves may play a role as well, but for practical reasons we leave them out of the discussion here. I found the following textbooks particularly useful: • Holton, J.R. An introduction to dynamic meteorology, Academic Press, Orlando, 1992. • Cushman-Roisin, B. Introduction to Geophyiscal Fluid Dynamics, Prentice- Hall, Englewood Cliffs, New Jersey, 1994. • Pedloski, J. Waves in the ocean and atmosphere, Introduction to wave dynamics, Springer, New York, 2003. • And for those who would like to see real Rossby waves in the ocean: D. B. Chelton and M. G. Schlax, Global Obeservations of Oceanic Rossby waves, Science 272, pp. 234-238 (1996) 2 Overview of wave types For all waves two balancing components are required. Inertia is relevant for every wave type, but the balancing component varies. A list of the main wave types and their common names are listed in Table 1. Waves that make use of the (small) compressibility of air or water are acous- tic or sound waves. However, acoustic waves have a too high wave speed to be relevant for geophysical flows. A second group of wave types are those who exist in barotropic flows, i.e. flows with uniform density. The incompressibility principle implies that the flow is non-divergent. Two steering forces are gravity and Coriolis. The simplest waves of this type are Kelvin waves, which will be analyzed below. Kelvin waves need a boundary, like the shore for the oceans or, strangely enough, the equator. Without boundaries, these type of waves are Poincar´ewaves. In fact, tsunamis are short Poincar´ewaves breaking on the coast, for which Coriolis forces are less important. Kelvin and Poincar´ewaves travel relatively fast. Rossby waves travel much slower, because they react on variations in the Coriolis force with the latitude. They have similarities with topographic waves, although these waves need a background flow. 2 Table 1: Overview of wave types Name Balancing term Essentially compressible Acoustic or sound waves Pressure variations Incompressible, barotropic Kelvin waves Gravity and Coriolis along a fluid boundary Poincar´ewaves Gravity and Coriolis Rossby waves (or planetary waves) Coriolis parameter variations Topograpic waves Flow depth variations Incompressible, baroclinic Internal (gravity / Buoyancy) waves Density differences In reality, the atmosphere and the ocean have a density that changes with height. That adds one other type of waves, namely internal waves. These waves are the least visible type of waves because these are completely inside the fluid and often are confined to narrow zones. 3 Analytical preparations 3.1 The linearized shallow water equations Here, we restrict ourselves to linear waves because they describe the dominant wave phenomena quite well. For simplicity, we start with a barotropic, one-layer system, thus the shallow water equations that have been derived in a previous lecture. We linearize these equations in order to find plain wave solutions. This simplicity is searched so that the physical mechanisms behind the wave dynamics can be understood. The shallow water equations are @u @u @u @ξ + u + v − fv = −g @t @x @y @x @v @v @v @ξ + u + v + fu = −g @t @x @y @y @h @uh @vh + + = 0 @t @x @y h = ξ + H in which we recall that ξ is the surface elevation, i.e. the elevation of the upper surface of the fluid. 3 We linearize the equations around a state of rest, i.e. a state with zero ve- locity, and zero surface elevation. The idea of linearization is that all variables (u, v, ξ) are assumed to have infinitesimal small amplitude. This means that terms linearly dependent on these variables have very small magnitude. How- ever, nonlinear terms, like the advection terms, e.g. uux have a magnitude that is even much and much smaller. So small in fact, that we can neglect them. We then arrive at the following set of equations: @u @ξ − fv = −g @t @x @v @ξ + fu = −g @t @y @h @u @v + H( + ) = 0 @t @x @y h = ξ + H These equations are the basis of what is to follow. 3.2 The reduced gravity model There are serious restrictions in applying the shallow-water equations to the real atmosphere and the real ocean. This is because fluids are treated as homo- geneous in the vertical, i.e. the density is assumed to be the same every where. In reality, stratification is one the key steering factors of oceanographic and atmospheric flows and cannot easily be neglected. Isn't it possible to build in some variation of density with height, so some stratification? The answer is yes. One can put a pile of shallow-water models on top of each other and connect them properly. The simplest of those models is the reduced gravity model (Figure 1). In the reduced gravity model the fluid is assumed to consist of two layers of different density, in which the lower layer fluid is at rest. To keep the lower layer at rest the horizontal pressure gradients have to be zero there. This introduces a special relation between the upper surface of the fluid and the interface between the two layers. It is shown below that the reduced gravity model has the same form as the shallow-water model, but the terms have a slightly different meaning. The reduced gravity model presented here is set up to represent the ocean, but with some reformulation such model can also represent the atmosphere. Then, density must be replaced by potential density and depth by height. Fur- thermore, the model can be extended to a multilayer reduced gravity model, in which still the lowest layer is at rest. We first integrate the pressure from the top of the fluid to somewhere in the lower layer, to level z, say. Using the hydrostatic equation we find: Z ξ Z ξ pz dz = − gρ dz z z The right-hand side of this equation can be evaluated by introducing the eleva- tion of the interface between the two layers, relative to a state of rest, as η. We 4 ξ Upper layer ρ1 η Lower layer ρ2 Figure 1: The reduced-gravity model, with surface elevation ξ and interface elevation η. Note that η is defined positive downward. choose η as positive downward, as depicted in Figure 1. We then find: Z ξ Z −(H+η) Z ξ − gρ dz = − gρ2 dz − gρ1 dz z z −(H+η) = −g(ξ + H + η)ρ1 − g(−(H + η) − z)ρ2 The left-hand side becomes: Z ξ pz dz = p(ξ) − p(z) = −p(z) z where we use that the pressure above the fluid p(ξ) is negligible. Hence the pressure at level z is given by: p(z) = g(ξ + H + η)ρ1 − g(z + H + η)ρ2 Our assumption is that the lower layer fluid is at rest, or that the horizontal pressure gradients are zero there. Taking the horizontal gradient of p(z) leads to: rp = gρ1rξ − g(ρ2 − ρ1)rη = 0 so that we find: gξ = g0η 0 0 in which g is the reduced gravity acceleration given by g = g(ρ2 − ρ1)/ρ1. Since the density difference between the layers is usually much smaller than the density itself we find that η >> ξ. This allows us to neglect ξ compared to η in the continuity equation, so that the set of equations describing the motion in the first layer becomes: 5 @u @η − fv = −g0 @t @x @v @η + fu = −g0 (1) @t @y @η @u @v + H( + ) = 0 @t @x @y where we used that ht = ηt, because H is constant. Notice that these equations for the reduced gravity model differ from those of the shallow-water equations in that g0 replaces g, and η replaces ξ. 3.3 Streamfunctions A useful property in quasi-geostrophic analysis is the streamfunction captur- ing the flow potential. It has some similarities with the surface interface height η, because η can also been seen as a flow potential. In the definition of , we use that the first order momentum balance is geostrophic: @η −f v = −g0 (2) 0 @x @η f u = −g0 0 @y If we eliminate the pressure gradient terms by cross differentiation (i.e. take the zonal derivative of the second and subtract the meridional derivative of the first) we find: f0(ux + vy) = 0 So, the horizontal divergence of the horizontal velocities is zero, as we have found before.