Numerical Weather Prediction 705 Much Larger-Scale Connected Structures with Complex Perovich, D

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 contributed 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

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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

V.18. Numerical Weather Prediction 707 to the design of stable and accurate algorithms for solv- 4 Solving the Equations ing the PDEs. Third, progress in meteorological theory, Analytical solution of the equations is impossible, so especially the development of the quasigeostrophic approximate methods must be employed. We consider equations and improved understanding of atmospheric methods of discretizing the spatial domain to reduce balance, provided a means to eliminate the spurious the PDEs to an algebraic system and of advancing the high-frequency oscillations that had spoiled Richard- solution in time. son’s forecast. Finally, the invention of high-speed digital computers enabled the enormous computational 4.1 Time-Stepping Schemes task of solving the equations to be undertaken. Let Q denote a typical dependent variable, governed by 3.2 The ENIAC Integrations an equation of the form dQ The first forecasts made using an automatic com- = F(Q). dt puter were completed in 1950 on the ENIAC (Elec- We replace the continuous-time domain t by a sequence tronic Numerical Integrator and Computer), the first of discrete times {0, Δt,2Δt,...,nΔt,...}, with the programmable general-purpose computer. The fore- solution at these times denoted by Qn = Q(nΔt). casts used a highly simplified model, representing the If this solution is known up to time t = nΔt, the atmosphere as a single layer and assuming conserva- right-hand term F n = F(Qn) can be computed. The tion of absolute vorticity expressed by the barotropic time derivative is now approximated by a centered vorticity equation, difference n+1 − n−1 d Q Q n (ζ + f)= 0, = F , dt 2Δt n+1 where ζ is the vorticity of the flow and f = 2Ω sin φ so the “forecast” value Q may be computed from n−1 n is the Coriolis parameter, with Ω the angular velocity the old value Q and the tendency F : + − of the Earth and φ the latitude. The Lagrangian time Qn 1 = Qn 1 + 2ΔtFn. derivative d ∂ This is called the leapfrog scheme. The process of step- = + V · ∇ dt ∂t ping forward from moment to moment is repeated a large number of times, until the desired forecast range includes the nonlinear advection by the flow. The equa- is reached. tion was approximated by finite differences in space The leapfrog scheme is limited by a stability criterion and time with a grid size of 736 km (at the North Pole) that restricts the size of the time step Δt. One way of and a time step of three hours. The resulting forecasts, circumventing this is to use an implicit scheme such as while far from perfect, were realistic and provided a + − − + powerful stimulus for further work. Qn 1 − Qn 1 F n 1 + F n 1 = . Baroclinic, or multilevel, models that enabled realis- 2Δt 2 tic representation of the vertical structure of the atmo- The time step is now unconstrained by stability, but the sphere were soon developed. Moreover, the simplified scheme requires the solution of the equation + + − − equations were replaced by more accurate primitive Qn 1 − ΔtFn 1 = Qn 1 + ΔtFn 1, equations, that is, the equations presented in section 2 which is prohibitive unless F(Q) is a linear function. but with the hydrostatic approximation. As these equa- Normally, implicit schemes are used only for particular tions simulate high-frequency gravity waves in addition (linear) terms of the equations. to the motions that are important for weather, the initial conditions must be carefully balanced. Techniques 4.2 Spatial Finite Differencing for ensuring this were developed. Most notable among these was the normal-mode initialization method: the For the PDEs that govern atmospheric dynamics we flow is resolved into normal modes and modified to must replace continuous variations in space by discrete ensure that the tendencies, or rates of change, of variables. The primary way to do this is to substitute the gravity wave components vanish. This suppresses finite-difference approximations for the spatial deriva- spurious oscillations. tives. It then transpires that the stability depends on

708 V. Modeling the relative sizes of the space and time steps. A real- field values Q(λ, φ) in the spatial domain. When the istic solution is not guaranteed by reducing their sizes model equations are transformed to spectral space they independently. become a coupled set of equations (ordinary differen- m We consider the simple one-dimensional wave equa- tial equations) for the spectral coefficients Qn . These tion are used to advance the coefficients in time, after which ∂Q ∂Q + c = 0, the new physical fields may be computed. ∂t ∂x In practice, the series expansion must be truncated where depends on both and , and where the Q(x, t) x t at some point: advection speed c is constant. We consider the sinu- N n − soidal solution Q = Q0eik(x ct) of wavelength L = = m m Q(λi,φj ,t) Qn (t)Yn (λi,φj ). 2π/k. We use centered difference approximations in n=0 m=−n both space and time: This is called triangular truncation, and the value of N n+1 n−1 n n Q − Q Q + − Q − indicates the resolution of the model. There is a compu- m m + c m 1 m 1 = 0, 2Δt 2Δx tational grid, called the Gaussian grid, corresponding to n = the spectral truncation. where Qm Q(mΔx,nΔt). We seek a solution of n = 0 ik(mΔx−CnΔt) the form Qm Q e . For real C, this is a wavelike solution. However, if C is complex, this solu- 5 Initial Conditions tion will behave exponentially, quite unlike the solution Numerical weather prediction is an initial-value prob- n of the continuous equation. Substituting Qm into the lem; to integrate the equations of motion we must spec- finite-difference equation, we find that ify the values of the dependent variables at an initial 1 − cΔt C = sin 1 sin kΔx . time. The numerical process then generates the val- kΔt Δx ues of these variables at later times. The initial data If the argument of the inverse sine is less than unity, C are ultimately derived from direct observations of the is real. Otherwise, C is complex, and the solution will atmosphere. grow with time. Thus, the condition for stability of the The optimal interpolation analysis method was, for solution is several decades, the most popular method of automatic cΔt 1. analysis for NWP. This method optimizes the combina- Δx tion of information in the background (forecast) field This is the Courant–Friedrichs–Lewy criterion, discov- and in the observations, using the statistical proper- ered in 1928. It imposes a strong constraint on the rel- ties of the forecast and observation errors to produce ative sizes of the space and time grids. The limitation an analysis that, in a precise statistical sense, is the best on stability can be circumvented by means of implicit possible analysis. finite differencing. Then An alternative approach to data assimilation is to find 2 − cΔt C = tan 1 sin kΔx . the analysis field that minimizes a cost function. This is kΔt 2Δx called variational assimilation and it is equivalent to the The numerical phase speed C is always real, so the statistical technique known as the maximum-likelihood implicit scheme is unconditionally stable, but the cost estimate, subject to the assumption of Gaussian errors. is that a linear system must be solved at each time step. When applied at a specific time, the method is called 4.3 Spectral Method three-dimensional variational assimilation, or 3D-Var for short. When the time dimension is also taken into In the spectral method, each field is expanded in a account, we have 4D-Var. series of spherical harmonics: ∞ n 5.1 Variational Assimilation = m m Q(λ, φ, t) Qn (t)Yn (λ, φ), n=0 m=−n The cost function for 3D-Var may be defined as the sum m of two components: where the coefficients Qn (t) depend only on time, and m where Yn (λ, φ) are the spherical harmonics J = JB + JO. m = imλ m Yn (λ, φ) e Pn (φ) We represent the model state by a high-dimensional m vector X. The term for longitude λ and latitude φ. The coefficients Qn of = 1 − T −1 − the harmonics provide an alternative to specifying the JB 2 (X XB) B (X XB)

V.18. Numerical Weather Prediction 709 represents the distance between the model state X and 5.2 Inclusion of the Time Dimension the background field X weighted by the background B Whereas conventional meteorological observations are error covariance matrix B. The term made at the main synoptic hours, satellite data are = 1 − T −1 − JO 2 (Y HX) R (Y HX) distributed continuously in time. To assimilate these represents the distance between the analysis and the data, it is necessary to perform the analysis over a observed values Y weighted by the observation error time interval rather than for a single moment. This is covariance matrix R. The observation operator H is a also more appropriate for observations that are dis- rectangular matrix that converts the background field tributed inhomogeneously in space. Four-dimensional into first-guess values of the observations. More gener- variational assimilation, or 4D-Var for short, uses all ally, the observation operator is nonlinear but, for ease the observations within an interval t0 t tN . The of description, we assume here that it is linear. cost function has a term JB measuring the distance to the background field X at the initial time t , just as in The minimum of J is attained at X = XA, where B 0 3D-Var. It also contains a summation of terms measur- ∇X J = 0, ing the distance to observations at each time step tn in that is, where the gradient of J with respect to each of the interval [t0,tN ]: the analyzed values is zero. Computing this gradient, N we get J = JB + JO(tn), n=0 −1 T −1 ∇X J = B (X − XB) + H R (Y − HX). where JB is defined as for 3D-Var and JO(tn) is given Setting this to zero we can deduce the expression by T −1 JO(tn) = (Yn − HnXn) R (Yn − HnXn). X = XB + K(Y − HXB). n The state vector X at time t is generated by integra- Thus, the analysis is obtained by adding to the back- n n tion of the forecast model from time t to t , written ground field a weighted sum of the difference between 0 n X =M (X ). The vector Y contains the observations observed and background values. The matrix K, the n n 0 n valid at time t . gain matrix, is given by n Just as the observation operator had to be linearized T T −1 K = BH (R + HBH ) . to obtain a quadratic cost function, we linearize the The analysis error covariance is then given by model operator Mn about the trajectory from the background field, obtaining what is called the tangent linear A = (I − KH)B. model operator Mn. Then we find that 4D-Var is for- The minimum of the cost function is found using mally similar to 3D-Var with the observation operator conjugate gradi- a descent algorithm such as the H replaced by HnMn. Just as the minimization of J ent method [IV.11 §4.1]; 3D-Var solves the minimiza- in 3D-Var involved the transpose of H, the minimization problem directly, avoiding computation of the gain tion in 4D-Var involves the transpose of HnMn, which T T T matrix. is MnHn. The operator Mn, the transpose of the tan- The 3D-Var method has enabled the direct assimi- gent linear model, is called the adjoint model. The con- lation of satellite radiance measurements. The error- trol variable for the minimization of the cost function prone inversion process, whereby temperatures are is X0, the model state at time t0, and the sequence of deduced from the radiances before assimilation, is thus analyses Xn satisfies the model equations, that is, the eliminated. Quality control of these data is also eas- model is used as a strong constraint. ier and more reliable. As a consequence, the accuracy The 4D-Var method finds initial conditions X0 such of forecasts has improved markedly since the intro- that the forecast best fits the observations within the duction of variational assimilation. The accuracy of assimilation interval. This removes an inherent disad- medium-range forecasts is now about equal for the two vantage of optimal interpolation and 3D-Var, where all hemispheres (see figure 1). This is due to better satel- observations within a fixed time window (typically of lite data assimilation. Satellite data are essential for the six hours) are assumed to be valid at the analysis time. Southern Hemisphere as conventional data are in such The introduction of 4D-Var at the European Centre short supply. The extraction of useful information from for Medium-Range Weather Forecasts (ECMWF) led to satellite soundings has been one of the great research a significant improvement in the quality of operational triumphs of NWP over the past forty years. medium-range forecasts.

710 V. Modeling

100

Day 3, Northern Hemisphere Day 3, Southern Hemisphere 90

80 Day 5, Northern Hem. Day 5, Southern Hemisphere

70 % 60 Day 7, Northern Hem. Day 7, Southern Hemisphere

50 Day 10, Northern Hemisphere 40 Day 10, Southern Hemisphere 30 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 2009 2011 2013

6 Forecasting Models Table 1 Upgrades to the ECMWF IFS in 2006 and 2010. The spectral resolution is indicated by the triangular trun- Operational forecasting today is based on output from cation number, and the effective resolution of the associ- ated Gaussian grid is indicated. The number of model lev- a suite of computer models. Global models are used for els, or layers used to represent the vertical structure of the predictions of several days ahead, while shorter-range atmosphere, is also given. forecasts are based on regional or limited-area models. Before After 2006 2006–9 2009 6.1 The ECMWF Global Model Spectral truncation T511 T799 T1279 As an example of a global model we consider the inte- Effective resolution 39 km 25 km 16 km grated forecast system (IFS) of the ECMWF (which is Model levels 60 91 137 based in Reading, in the United Kingdom). The ECMWF produces a wide range of global atmospheric and terminated at total wave number 1279. This is equiv- marine forecasts and disseminates them on a regu- alent to a spatial resolution of 16 km. The number of lar schedule to its thirty-four member and cooperat- model levels in the vertical has recently been increased ing states. The primary products are deterministic fore- to 137. The new Gaussian grid for the IFS has about casts for the atmosphere out to ten days ahead, based 2 × 106 points. With 137 levels and five primary prog- on a high-resolution model, and probabilistic forecasts, nostic variables at each point, about 1.2 × 109 num- extending to a month, made using a reduced resolution bers are required to specify the atmospheric state at and an ensemble of fifty-one model runs. a given time. That is, the model has about a billion The basis of the NWP operations at the ECMWF is the degrees of freedom. The computational task of making IFS. It uses a spectral representation of the meteoro- forecasts with such high resolution is truly formidable. logical fields. The IFS system underwent major resolu- The ECMWF carries out its operational program using tion upgrades in 2006 and in 2010. Table 1 compares a powerful and complex computer system. At the heart the spatial resolutions of the three model cycles, indi- of this system is a Cray XC30 high-performance com- cating the substantial improvements in model resolu- puter, comprising some 160 000 processors, with a sus- tion in recent years. The truncation of the deterministic tained performance of over 200 teraflops (2 × 1014 model is now T1279; that is, the spectral expansion is floating-point operations per second).

V.18. Numerical Weather Prediction 711

6.2 Mesoscale Modeling of the deterministic forecast. Probability forecasts for a wide range of weather events are generated and dissem- Short-range forecasting requires detailed guidance that inated for use in the operational centers, and they have is updated frequently. Many national meteorological become the key tools for medium-range prediction. services run limited-area models with high resolution to provide such forecast guidance. These models per- 7 Verification of ECMWF Forecasts mit a free choice of geographical area and spatial resolution, and forecasts can be run as frequently as Forecast accuracy has improved dramatically in recent required. Limited-area models make available a com- decades. This can be measured by the anomaly corre- prehensive range of outputs, with a high time resolu- lation. The anomaly is the difference between a fore- tion. Nested grids with successively higher resolution cast value and the corresponding climate value, and can be used to provide greater local detail. the agreement between the forecast anomaly and the The Weather Research and Forecasting Model is a observed anomaly is expressed as the anomaly correla- next-generation mesoscale NWP system developed in a tion. The higher this score the better; by general agree- partnership involving American national agencies (the ment, values in excess of 60% imply skill in the forecast. National Centers for Environmental Prediction and the In figure 1, the twelve-month running mean anomaly National Center for Atmospheric Research) and uni- correlations (in percentages) of the three-, five-, seven-, versities. It is designed to serve the needs of both and ten-day 500 hPa height forecasts are shown for operational forecasting and atmospheric research. The the extratropical Northern Hemisphere and Southern Weather Research and Forecasting Model is suitable for Hemisphere. The lines above each shaded region are a broad range of applications, from meters to thou- for the Northern Hemisphere and the lines below are sands of kilometers, and it is currently in operational for the Southern Hemisphere, with the shading showing use at several national meteorological services.1 the difference in scores between the two. The plots in figure 1 show a continuing improvement 6.3 Ensemble Prediction in forecast accuracy, especially for the Southern Hemi- The chaotic nature of atmospheric flow is now well sphere. By the turn of the millennium, the accuracy was understood. It imposes a limit on predictability, as comparable for the two hemispheres. Predictive abil- unavoidable errors in the initial state grow rapidly and ity has improved steadily over the past thirty years, render the forecast useless after some days. As a result and there is now accuracy out to eight days ahead. of our increased understanding of the inherent diffi- This record is confirmed by a wealth of other data. Pre- culties of making precise predictions, there has been dictive skill has been increasing by about one day per a paradigm shift in recent years from deterministic decade, and there are reasons to hope that this trend to probabilistic prediction. A forecast is now consid- will continue for several more decades. ered incomplete without an accompanying error bar, or quantitative indication of confidence. 8 Applications of NWP The most successful way of producing a probabilis- NWP models are used for a wide range of applica- tic prediction is to run a series, or ensemble, of fore- tions. Perhaps the most important purpose is to pro- casts, each starting from a slightly different initial state vide timely warnings about weather extremes. Great and each randomly perturbed during the forecast to financial losses can be caused by gales, floods, and simulate model errors. The ensemble of forecasts is other anomalous weather events. The warnings that used to deduce probabilistic information about future result from NWP guidance can greatly diminish losses changes in the atmosphere. Since the early 1990s this of both life and property. Transportation, energy con- systematic method of providing an a priori measure sumption, construction, tourism, and agriculture are of forecast accuracy has been operational at both the all sensitive to weather conditions. There are expec- ECMWF and at the National Centers for Environmen- tations from all these sectors of increasing accu- tal Prediction in Washington. In the ECMWF’s ensem- racy and detail in short-range forecasts, as decisions ble prediction system, an ensemble of fifty-one fore- with heavy financial implications must continually be casts is performed, each having a resolution half that made. NWP models are used to generate special guidance 1. Full details of the system are available at www.wrf-model.org. for the marine community. Predicted winds are used to

712 V. Modeling drive wave models, which predict sea and swell heights Further Reading and periods. Prediction of road ice is performed by Lynch, P. 2006. The Emergence of Numerical Weather Pre- specially designed models that use forecasts of tem- diction: Richardson’s Dream. Cambridge: Cambridge Uni- perature, humidity, precipitation, cloudiness, and other versity Press. parameters to estimate the conditions on the road surface. Trajectories are easily derived from limited-area models. These are vital for modeling pollution drift, for V.19 Tsunami Modeling nuclear fallout, smoke from forest fires, and so on. Avi- Randall J. LeVeque ation benefits significantly from NWP guidance, which provides warnings of hazards such as lightning, icing, 1 Introduction and clear-air turbulence. The general public’s appreciation of the danger of tsunamis has soared since the Indian Ocean tsunami 9 The Future of December 26, 2004, killed more than 200 000 peo- ple. Several other large tsunamis have occurred since Progress in NWP over the past sixty years can be accu- then, including the devastating March 11, 2011, Great rately described as revolutionary. Weather forecasts Tohoku tsunami generated off the coast of Japan. The are now consistently accurate and readily available. international community of tsunami scientists has also Nevertheless, some formidable challenges remain. Sud- grown considerably since 2004, and an increasing num- den weather changes and extremes cause much human ber of applied mathematicians have contributed to the hardship and damage to property. These rapid devel- development of better models and computational tools opments often involve intricate interactions between for the study of tsunamis. In addition to its impor- dynamical and physical processes, both of which vary tance in scientific studies and public safety, tsunami on a range of timescales. The effective computational modeling also provides an excellent case study to illus- coupling between the dynamical processes and the trate a variety of techniques from applied and com- physical parametrizations is a significant challenge. putational mathematics. This article combines a brief Nowcasting is the process of predicting changes over overview of tsunami science and hazard mitigation periods of a few hours. Guidance provided by current with descriptions of some of these mathematical tech- numerical models occasionally falls short of what is niques, including an indication of some challenging required to take effective action and avert disasters. problems of ongoing research. Greatest value is obtained by a systematic combina- The term “tsunami” is generally used to refer to any tion of NWP products with conventional observations, large-scale anomalous motion of water that propagates radar imagery, satellite imagery, and other data. But as a wave in a sizable body of water. Tsunamis differ much remains to be done to develop optimal now- from familiar surface waves in several ways. Typically, casting systems, and we may be optimistic that future the fluid motion is not confined to a thin layer of water developments will lead to great improvements in this near the surface, as it is in wind-generated waves. Also area. the wavelength of the waves is much longer than nor- At the opposite end of the timescale, the chaotic mal: sometimes hundreds of kilometers. This is orders nature of the atmosphere limits the validity of deter- of magnitude larger than the depth of the ocean (which ministic forecasts. Interaction between the atmosphere is about 4000 m on average), and tsunamis are there- and the ocean becomes a dominant factor at longer fore also sometimes referred to as “long waves” in the forecast ranges, as does coupling to sea ice [V.17]. scientific literature. In the past, tsunamis were often Also, a more accurate description of aerosols and trace called “tidal waves” in English because they share some gases should improve long-range forecasts. Although characteristics with tides, which are the visible effect good progress in seasonal forecasting for the tropics of very long waves propagating around the Earth. How- has been made, the production of useful long-range ever, tsunamis have nothing to do with the gravitational forecasts for temperate regions remains to be tack- (tidal) forcing that drives the tides, and so this term is led by future modelers. Another great challenge is the misleading and is no longer used. The Japanese word modeling and prediction of climate change, a matter of “tsunami” means “harbor wave,” apparently because increasing importance and concern. sailors would sometimes return home to find their

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