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1 Theory of Tropical Moist Convection OUP UNCORRECTED PROOF – FIRST PROOF, 30/10/2019, SPi 1 Theory of tropical moist convection David M. Romps Department of Earth and Planetary Science, University of California, Berkeley, CA Climate and Ecosystem Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA Romps, D., Theory of tropical moist convection. In: Fundamental aspects of turbulent flows in climate dynamics. Edited by Freddy Bouchet, Tapio Schneider, Antoine Venaille, Christophe Salomon: Oxford University Press (2020). © Oxford University Press. DOI: 10.1093/oso/9780198855217.003.0001 1 Theory of Tropical Moist Convection David M. Romps Department of Earth and Planetary Science, University of California, Berkeley, CA Climate and Ecosystem Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 1.1 Introduction These lecture notes cover the theory of tropical moist convection. Many simplifications are made along the way, like neglecting rotation and treating the atmosphere as a two- dimensional fluid or even reducing the atmosphere to two columns. We can gain an immense amount of insight into the real atmosphere by studying these toy models, including answers to the following questions: what is the dominant energy balance in the tropical free troposphere; what sets the temperature structure of the tropical free troposphere; what happens to the pulse of heating deposited into the atmosphere by a rain cloud; why does the tropical atmosphere have the relative-humidity profile that it does; and what sets the amount of energy available to storms? These notes attempt to give the first-order answers to these questions in a format that is accessible to beginning graduate students. We begin with a discussion of atmospheric energy. 1.2 Dry thermodynamic equations Fluids are governed by conservation laws: conservation of mass, conservation of energy, and conservation of momentum. Each of these conservation laws, when written down as an equation, prognoses (i.e., predicts or governs) the evolution of the fluid's mass, energy, and momentum, respectively. Sometimes, we write down those governing equa- tions in terms of closely related variables. For example, we might rewrite the equation for energy in terms of an equation for temperature T . Or, we might rewrite an equa- tion for momentum as an equation for velocity ~u. But, no matter how the equations may be written in the end, they are all derivable from equations that plainly state the conservation of mass, energy, and momentum. The best way to derive these conservation equations is to start in Eulerian form. An equation in Eulerian form has a term that is an Eulerian time derivative of some quantity. An Eulerian time derivative is just a partial derivative with respect to time, Dry thermodynamic equations 3 Lx Ly z Lz y x Fig. 1.1 Imaginary box, fixed in space. You may think of this box being infinitesimally small. We derive equations describing the mean quantities of the fluid in this box, as the fluid passes through this box. Eventually, we take the limits of Lx, Ly and Lz to zero, such that the box reduces to a single fixed point in space. @=@t. (It may seem silly to give @=@t a special name like \Eulerian time derivative", but this is necessary to distinguish it from the \Lagrangian time derivative" d=dt, which is also commonly used in the study of fluids.) In Eulerian form, we can write any conservation law as Storage rate of X in the box + Export rate of X out of the box = Sources of X in the box ; (1.1) where X is either mass, momentum in one of the three independent directions, or energy. Here, the \box" is some box-shaped region { perhaps a cube { that has a fixed size, shape, orientation, and position with respect to the chosen coordinate system (typically, x, y, and z). It is most important that this box be stationary: the fluid can and will pass through the box, but the box that we consider must not move. For simplicity, we will imagine a box whose edges are parallel to the x, y, and z axes and that have lengths Lx, Ly, and Lz, respectively; see Figure 1.1. In the equation above, the \storage rate" or \tendency" is the rate at which the amount of X in the box is increasing. The \export rate" or \divergence" is the rate at which X is being carried by the fluid out of the box by passing through the walls of the box. The \sources" are any addition of X to the box (or, if negative, subtraction of X from the box) through means other than being carried into or out of the box by the fluid flow. To write out equation (1.1) in more detail, note that the amount of X in the box 1 is equal to the volume of the box V times the density of X. So, the storage rate is @t of the volume of the box V times the density of X. But, since V is constant in time, Storage rate of X in the box = V@t(density of X) : The export rate of X is equal to the net rate at which X is carried outwards across the faces of the box. Let us consider the two faces whose normals are aligned with the 1Here and throughout, the partial derivatives @=@t, @=@x, @=@y, and @=@z will be abbreviated as @t, @x, @y, and @z. This is simply a matter of convenience: it reduces clutter and saves ink. 4 Moist Convection x axis. The rate at which X passes through one of those faces is equal to the area of 2 the face LyLz times u times the density of X. The net export across the two faces withx ^ normals is then h i L L u × X j − u × X j ; ( y z) ( density of ) +Lx=2 ( density of ) −Lx=2 where the subscripts −Lx=2 and +Lx=2 indicate that the expressions should be eval- uated at the location of these faces, which are at positions −Lx=2 and +Lx=2 relative to the box's center. Now, if we imagine that our box is small compared to the spatial variations in u or the density of X, then we can write this as " u × X j − u × X j # ( density of ) +Lx=2 ( density of ) −Lx=2 (LyLz) Lx Lx = (LyLz)[Lx@x(u × density of X)] = V@x(u × density of X) : Similarly, the net exports through the twoy ^-normal and twoz ^-normal faces are V@y(v × density of X) and V@z(w × density of X) ; respectively. Altogether, the net amount of X being exported through the faces of the box is Export rate of X out of box = V r~ · (~u × density of X) : The only thing left to figure out, then, is the set of sources for X. For mass, there is none: Sources of mass to the box = 0 : Although the diffusive fluxes of mass, momentum, and energy can be important, es- pecially near the surface, we will ignore diffusion for simplicity. For momentum, there are gravity and pressure forces: Sources of momentum to the box = V ~g − V r~ p : Here, ~g = (0; 0; −9:81 m s−2) is the gravitational acceleration vector and p is the fluid pressure. Noting that the pressure of a fluid is isotropic (the same in all directions) and recalling that pressure is simply force per area, the pressure-gradient term −V r~ p in the equation above can be derived by considering the forces on the faces of the box. For energy (by which I mean the sum of internal energy, kinetic energy, and gravitational potential energy), the sources that we need to worry about are radiation and pressure work: Sources of energy to the box = VQ − V r~ · (p~u) : Here, Q is the net radiative heating per volume with units of W m−3 and the last term represents the energy added to the box by being pushed by adjacent fluid at the faces 2Here, we adopt the standard meteorological convention of referring to the three components of the fluid velocity by u, v, and w, corresponding to x, y, and z axes, respectively. Dry thermodynamic equations 5 of the box. Like the pressure-gradient term in the momentum equation, the −V r·~ (p~u) term can be derived by considering the force-times-distance work performed on the parcel of air at the faces of the box. Putting it all together, denoting the density of mass, momentum, and energy by ρ, ρ~u, and ρE, and dividing by the constant V , we get @tρ + r~ · (ρ~u) = 0 (1.2) @t(ρ~u) + r~ · (ρ~u~u) = ρ~g − r~ p (1.3) @t(ρE) + r~ · (ρE~u) = Q − r~ · (p~u) ; (1.4) where E is the specific3 energy of the fluid and the pressure is specified by the ideal gas law to be p = RρT , where R is the specific gas constant of air. The energy equation (a.k.a., the thermodynamic equation) plays a prominent role in these lectures because energy is tightly connected to the topics of buoyancy and atmospheric convection. Although there are many different \energy" or \thermody- namic" variables for dry air, they are all functions of4 temperature T and also, possibly, pressure p or height z. Therefore, we can write a general thermodynamic variable for dry air as f(T; p; z). At a given height, the buoyancy b of a dry parcel in a dry atmo- sphere is defined in terms of the temperature of the parcel Tp and the temperature of the environment at the same height Te as T − T b = g p e ; (1.5) Te where g ≈ 10 m s−2 is the magnitude of the gravitational acceleration.
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