© Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. V.18. Numerical Weather Prediction 705 much larger-scale connected structures with complex Perovich, D. K., J. A. Richter-Menge, K. F. Jones, and B. Light. boundaries. Connectivity of melt ponds promotes fur- 2008. Sunlight, water, and ice: extreme Arctic sea ice melt ther melting and the breakup of floes, as well as hor- during the summer of 2007. Geophysical Research Letters izontal transport of meltwater and drainage through 35:L11501. Stauffer, D., and A. Aharony. 1992. Introduction to Percola- cracks, leads, and seal holes. tion Theory, 2nd edn. London: Taylor & Francis. Acknowledgments. I gratefully acknowledge support from Stroeve, J., M. M. Holland, W. Meier, T. Scambos, and M. Ser- the Division of Mathematical Sciences and the Division of reze. 2007. Arctic sea ice decline: faster than forecast. Polar Programs at the U.S. National Science Foundation Geophysical Research Letters 34:L09501. (NSF) through grants DMS-1009704, ARC-0934721, DMS- Strong, C., and I. G. Rigor. 2013. Arctic marginal ice zone 0940249, and DMS-1413454. I am also grateful for sup- trending wider in summer and narrower in winter. Geo- port from the Arctic and Global Prediction Program and physical Research Letters 40(18):4864–68. the Applied Computational Analysis Program at the Office Thomas, D. N., and G. S. Dieckmann, editors. 2009. Sea Ice, of Naval Research through grants N00014-13-10291 and 2nd edn. Oxford: Wiley-Blackwell. N00014-12-10861. Finally, I would like to thank the NSF Thompson, C. J. 1988. Classical Equilibrium Statistical Math Climate Research Network for their support of this Mechanics. Oxford: Oxford University Press. work, as well as many colleagues and students who con- tributed so much to the research represented here, espe- Torquato, S. 2002. Random Heterogeneous Materials: Micro- cially Steve Ackley, Hajo Eicken, Don Perovich, Tony Worby, structure and Macroscopic Properties. New York: Springer. Court Strong, Elena Cherkaev, Jingyi Zhu, Adam Gully, Ben Untersteiner, N. 1986. The Geophysics of Sea Ice. New York: Murphy, and Christian Sampson. Plenum. Washington, W., and C. L. Parkinson. 2005. An Introduc- Further Reading tion to Three-Dimensional Climate Modeling, 2nd edn. Avellaneda, M., and A. Majda. 1989. Stieltjes integral rep- Herndon, VA: University Science Books. resentation and effective diffusivity bounds for turbulent transport. Physical Review Letters 62:753–55. Bergman, D. J., and D. Stroud. 1992. Physical properties V.18 Numerical Weather Prediction of macroscopically inhomogeneous media. Solid State Peter Lynch Physics 46:147–269. Cherkaev, E. 2001. Inverse homogenization for evaluation 1 Introduction of effective properties of a mixture. Inverse Problems 17: 1203–18. The development of computer models for numeri- Eisenman, I., and J. S. Wettlaufer. 2009. Nonlinear threshold cal simulation and prediction of the atmosphere and behavior during the loss of Arctic sea ice. Proceedings of oceans is one of the great scientific triumphs of the the National Academy of Sciences of the USA 106(1):28–32. Feltham, D. L. 2008. Sea ice rheology. Annual Review of Fluid past fifty years. Today, numerical weather prediction Mechanics 40:91–112. (NWP) plays a central and essential role in operational Flocco, D., D. L. Feltham, and A. K. Turner. 2010. Incorpora- weather forecasting, with forecasts now having accu- tion of a physically based melt pond scheme into the sea racy at ranges beyond a week. There are several rea- ice component of a climate model. Journal of Geophysical sons for this: enhancements in model resolution, better Research 115:C08012. numerical schemes, more realistic parametrizations of Golden, K. M. 2009. Climate change and the mathematics of physical processes, new observational data from satel- transport in sea ice. Notices of the American Mathematical Society 56(5):562–84 (and issue cover). lites, and more sophisticated methods of determining Hohenegger, C., B. Alali, K. R. Steffen, D. K. Perovich, and the initial conditions. In this article we focus on the fun- K. M. Golden. 2012. Transition in the fractal geometry of damental equations, the formulation of the numerical Arctic melt ponds. The Cryosphere 6:1157–62. algorithms, and the variational approach to data assim- Hunke, E. C., and W. H. Lipscomb. 2010. CICE: the Los ilation. We present the mathematical principles of NWP Alamos sea ice model. Documentation and Software and illustrate the process by considering some specific User’s Manual, version 4.1. LA-CC-06-012, t-3 Fluid Dy- models and their application to practical forecasting. namics Group, Los Alamos National Laboratory. Milton, G. W. 2002. Theory of Composites. Cambridge: Cam- bridge University Press. 2 The Basic Equations Orum, C., E. Cherkaev, and K. M. Golden. 2012. Recovery of inclusion separations in strongly heterogeneous compos- The atmosphere is governed by the fundamental laws ites from effective property measurements. Proceedings of physics, expressed in terms of mathematical equa- of the Royal Society of London A 468:784–809. tions. They form a system of coupled nonlinear partial For general queries, contact [email protected] © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. 706 V. Modeling differential equations (PDEs). These equations can be given, the above system of seven (scalar) equations pro- used to predict the evolution of the atmosphere and to vides a complete description of the evolution of the simulate its long-term behavior. seven variables {u, v, w, p, ρ, T , q}. The primary variables are the fluid velocity V (with For large-scale motions the vertical component of three components, u eastward, v northward, and w velocity is very much smaller than the horizontal com- upward), pressure p, density ρ, temperature T , and ponents, and we can replace the vertical equation by humidity q. Using Newton’s laws of motion and the a balance between the vertical pressure gradient and principles of conservation of energy and mass, we can gravity. This yields the hydrostatic equation obtain a system whose solution is well determined by ∂p + gρ = 0. the initial conditions. ∂z The central components of the system, govern- Hydrostatic models were used for the first fifty years of ing fluid motion, are the navier–stokes equations NWP but nonhydrostatic models are now coming into [III.23]. We write them in vector form: widespread use. ∂V 1 + V · ∇V + 2Ω × V + ∇p = F + g. 3 The Emergence of NWP ∂t ρ The equations are relative to the rotating Earth and The idea of calculating the changes in the weather Ω is the Earth’s angular velocity. In order, the terms by numerical methods emerged around the turn of of this equation represent local acceleration, nonlin- the twentieth century. Cleveland Abbe, an American ear advection, Coriolis term, pressure gradient, friction, meteorologist, viewed weather forecasting as an appli- and gravity. The friction term F is small in the free cation of hydrodynamics and thermodynamics to the atmosphere but is crucially important in the boundary atmosphere. He also identified a system of mathemat- layer (roughly, the first 1 km above the Earth’s surface). ical equations, essentially those presented in section 2 The apparent gravity g includes the centrifugal force, above, that govern the evolution of the atmosphere. which depends only on position. This idea was developed in greater detail by the Norwe- The temperature, pressure, and density are linked gian Vilhelm Bjerknes, whose stated goal was to make through the equation of state meteorology an exact science: a true physics of the atmosphere. p = RρT, 3.1 Richardson’s Forecast where R is the gas constant for dry air. In practice, a slight elaboration of this is used that takes account of During World War I, Lewis Fry Richardson, an English moisture in the atmosphere. Quaker mathematician, calculated the changes in the Energy conservation is embodied in the first law of weather variables directly from the fundamental equa- thermodynamics, tions and presented his results in a book, Weather dT Prediction by Numerical Process, in 1922. His predic- cv + RT∇ · V = Q, dt tion of pressure changes was utterly unrealistic, being two orders of magnitude too large. The primary cause where cv is the specific heat at constant volume and of this failure was the inaccuracy and imbalance of Q is the diabatic heating rate. Conservation of mass is the initial conditions. Despite the outlandish results, expressed in terms of the continuity equation: Richardson’s methodology was unimpeachable, and is dρ + ρ∇ · V = 0. essentially the approach we use today to integrate the dt equations. Finally, conservation of water substance is expressed Richardson was several decades ahead of his time. by the equation For computational weather forecasting to become a dq = S, practical reality, advances on a number of fronts were dt required. First, an observing system for the tropo- where q is the specific humidity and S represents all sphere, the lowest layer of the atmosphere, extending sources and sinks of water vapor. to about 12 km, was established to serve the needs of Once initial conditions, appropriate boundary con- aviation; this also provided the initial data for weather ditions, and external forcings, sources, and sinks are forecasting. Second, advances in numerical analysis led For general queries, contact [email protected] © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. V.18. Numerical Weather Prediction 707 to the design of stable and accurate algorithms for solv- 4 Solving the Equations ing the PDEs.