## 1. Introduction

In this study the Jacobian-free Newton–Krylov (JFNK) approach will be used to solve an implicitly balanced version of a physical model consisting of the Navier–Stokes equations with additional equations representing cloud processes. As discussed in Knoll et al. (2003), implicitly balanced refers to function evaluations occurring at the same instant in time, thus implying for the Crank–Nicolson solution procedure that all forcing terms in the physical model are computed at time level *n* + 1/2. In this paper, time splitting will be defined as not being implicitly balanced or a solution procedure employing function evaluations at differing time levels. Currently, most nonhydrostatic models used to simulate hurricanes (Davis and Bosart 2002) employ time splitting in their construction and thus are not implicitly balanced. A key purpose of the hurricane simulations to be presented in this paper will be to demonstrate that, by employing a time-stepping procedure that is implicitly balanced, highly time accurate solutions can be obtained more efficiently than from a time-split solution procedure.

If all forcing terms within the physical model are computed at time level *n* + 1/2, then a nonlinear solver such as JFNK must be used (Reisner et al. 2003); however, if only a select few terms are computed at new time levels, then traditional time-split approaches used in atmospheric science such as the semi-implicit (Benoit et al. 1997) or split-explicit (Wicker and Skamarock 2002) can be formulated from the physical model. For instance, the semi-implicit solution procedure usually involves substituting the momentum equations into a pressure or continuity equation and then implicitly solving the resulting equation (Harlow and Amsden 1971; Skamarock et al. 1997) with only terms involving pressure being computed at the new time level. In this paper, the semi-implicit approach will be defined by the action of forming a single linear implicit equation for pressure by substituting the momentum equations into the energy equation present in the physical model. A positive aspect of the semi-implicit approach is that it enables slower processes such as advection to use time steps larger than the sound wave Courant–Friedrichs–Lewy (CFL) number, for example, Δ*t*_{s} = min(Δ*x*,Δ*y*,Δ*z*)/ *c*_{sound}, where *c*_{sound} is the speed of sound. Hence, efficiency gains come from not computing slower physical processes as frequently as would be required by the sound wave CFL. But, since the semi-implicit algorithm is linear, nonlinear terms are usually approximated by a first-order-in-time Taylor series expansion and thus the semi-implicit method should be less accurate (e.g., first-order-in-time) for the current physical model than a second-order-in-time implicitly balanced nonlinear solution procedure.

The chosen nonlinear solver employs the JFNK method that is a combination of Newton’s method and an iterative Krylov method used in solving the linear system formed for each Newton iteration (Brown and Saad 1990; Chan and Jackson 1984). Because the JFNK method does not involve the formation of individual Jacobian elements during the solution of the linear system, the algorithm is said to be “Jacobian free.” But, in practice, the method is not fully “matrix free” since matrices are usually formed within the preconditioner (Reisner et al. 2001). For time step sizes below the dynamical time scale of the given problem, the fully implicit JFNK solution algorithm combined with a Crank–Nicolson (e.g., trapezoidal) time discretization enables the numerical procedure to achieve second-order-in-time accuracy (Mousseau et al. 2002; Reisner et al. 2003). The second-order-in-time accuracy, combined with a robust physics-based preconditioner, has enabled the JFNK solver to achieve a given level of temporal error in a shorter amount of CPU time as compared to a semi-implicit algorithm for a rotating shallow-water system (Mousseau et al. 2002) and for a dry bubble (Reisner et al. 2003). The JFNK approach also enables time step sizes that are larger than the time scales associated with processes such as advection, gravity waves, and sound waves. One of the goals of this paper will be to demonstrate that the positive numerical behavior of the implicitly balanced approach, demonstrated in the previous simpler systems, still holds in a more complex system represented by a hurricane.

The physics-based preconditioning process is defined by employing another solution algorithm as a preconditioner for the linear system found within the JFNK method. The chosen solution algorithm was the semi-implicit method. Previous work employing a physics-based preconditioner of this form has shown that by efficiently removing the fastest time scales found in the system, the number of Krylov iterations were significantly reduced (Mousseau et al. 2002; Reisner et al. 2003), hence increasing the overall efficiency of the JFNK approach. Another positive aspect of the physics-based preconditioner is the ability of the entire algorithm to significantly reduce the memory allocation of a given calculation. This reduction becomes especially crucial for large three-dimensional problems that are being inverted by iterative Krylov solvers such as the Generalized Minimum Residual GMRES (Saad 1996), which require storage of previous Krylov vectors.

Because the physics-based preconditioner employs a semi-implicit solution procedure and hence is also a solution method to the physical model, it will be one of the time-split solution methods that will be compared against the implicitly balanced algorithm used in the subsequent simulations; however, unlike most traditional semi-implicit approaches used in atmospheric science that employ leapfrog time stepping, the physics-based preconditioner employs a first-order-in-time numerical approach. To quantify differences between the first-order semi-implicit approach against the traditional leapfrog semi-implicit approach, the two approaches will be compared against the implicitly balanced method in simulations of a moist bubble.

Unlike the complex bulk microphysical models typically employed in a majority of mesoscale models (e.g., Reisner et al. 1998), a simple and differentiable parameterization that converts water vapor into total cloud substance was utilized for this paper. The microphysical model is completed by the addition of the traditional bulk parameterization of the falling of rain. It is not the point of this paper to promote the utility of such a simple microphysical model nor demonstrate its ability to successfully reproduce an observed convective event. The purpose of this paper is to reveal the impact that time splitting this microphysical model has on the accuracy of idealized moist-bubble and hurricane simulations. The use of a highly idealized microphysical model to simulate clouds within a hurricane is not without precedent (e.g., Rotunno and Emanuel 1987).

The remainder of the paper is organized as follows: in the next section the physical model and its discretization are presented; the third section provides a brief description of the JFNK approach; in the fourth section results that show the accuracy of the implicitly balanced model versus semi-implicit models for simulating an idealized moist precipitating bubble are shown; in the fifth section hurricane simulations utilizing the implicitly balanced model are compared against results obtained from the first-order semi-implicit model; and the final section offers a few concluding remarks and future directions. Details concerning the embedding of a preconditioner within the JFNK approach and the physics-based preconditioner itself are presented in appendix A. The leapfrog semi-implicit approach used in the moist-bubble simulations is discussed in appendix B.

## 2. Compressible model

*u*and

*υ*are the Cartesian velocities in the horizontal,

*x*and

*y,*directions;

*w*is the Cartesian velocity in the vertical,

*z*, direction;

*f*= 2Ω sin

*φ*and

*f̃*= 2Ω cos

*φ*are the

*z*and

*y*components of the earth’s rotation axis at the latitude

*φ*;

*p*is the pressure of the gas;

*ρ*is the total density of the air,

*ρ*=

*ρ*+

_{d}*ρ*with

_{υ}*ρ*the dry air density and

_{d}*ρ*the density of water vapor;

_{υ}*g*is the acceleration due to gravity;

*p*′ =

*p*−

*p*is the pressure perturbation with

_{e}*p*=

_{e}*p*(

_{e}*z*) the environmental pressure;

*ρ*′ =

*ρ*−

*ρ*is the density perturbation where

_{e}*ρ*=

_{e}*ρ*(

_{e}*z*) is the environmental density;

*u*and

_{e}*υ*are the environmental winds; and

_{e}

*τ*^{i′j′}= (∂

*u*/∂

^{i′}*x*) + (∂

^{j′}*u*/∂

^{j′}*x*) − (⅔)

^{i′}*δ*

^{i′j′}(∂

*u*/∂

^{s′}*x*) is the strain-rate tensor with the indices,

^{s′}*i*′,

*j*′, and

*s*′ ranging from 1 to 3.

*θ*is the potential temperature,

*θ*=

*T*(

*p*

_{o}/

*p*)

^{Rd/Cp}, with

*T*the temperature of the gas and the diffusional flux of potential temperature being defined, for example, in the

*x*direction as

*F*

_{θx}=

*ρ*

*κ*(∂

*θ*/∂

*x*), with

*κ*being the coefficient of diffusion. Note constants such as

*C*are defined in Table 1.

_{p}*f*

_{cloud},represents a conversion term from water vapor,

*q*, to total cloud substance,

_{υ}*q*, and the subsequent energy release where qvs = 0.622[(esw/

_{c}*p*− esw)] is the saturated vapor mixing ratio, with esw = 6.112 exp{17.67[(

*T*−

*T*

_{f})/(

*T*− 29.65)]} for

*T*>

*T*or esw = 6.112 exp(22.514 − 6150/

_{f}*T*) for

*T*<

*T*. The constant, Φ

_{f}*, appearing in Eq. (5) was chosen so the time discretization resolved the conversion of*

_{c}*q*into

_{υ}*q*for a specified grid resolution, with the moist-bubble simulations employing Φ

_{c}*= 0.25 and the hurricane simulations utilizing Φ*

_{c}*= 1.5 × 10*

_{c}^{−3}. The termrepresents a flux of energy from the ocean’s surface.

*q*,

_{υ}*q*, and

_{c}*ρ*are expressed aswith a diffusional flux of a given scalar being defined by the same procedure as was done in the energy equation. The term,

*f*

_{fall}, appearing in Eq. (8), represents the falling of

*q*(see Reisner et al. 1998). The parameterization was computed fromwhere

_{c}*V*

_{f}= (

*a*

_{qc}Γ(4 +

*b*

_{qc})/6

*λ*

^{bqc}), with

*λ*= [

*π*

*ρ*

_{qc}

*N*

_{qc}/(

*ρ*

*q*

_{c})]. The term,

*f*

_{surface−gas}, represents the flux of

*q*from the ocean’s surface, and in analytical form is expressed in a form similar to Eq. (6).

_{υ}*C*

_{o}=

*R*

^{Γ}

_{d}/

*p*

_{o}

^{Rd/Cυ}with Γ =

*C*

_{p}/

*C*

_{υ}. The discrete equations employ a Crank–Nicolson time discretization on a rectangular Cartesian coincident or nonstaggered grid. Employing implied spatial subscripts of

*i*−

*x*,

*j*−

*y*, and

*k*−

*z*where only exceptions are given, Eqs. (1)–(11) can be written in discretized form aswhere

*i*∈ [1,

*nx*],

*j*∈ [1,

*ny*], and

*k*∈ [1,

*nz*], with

*nx, ny,*and

*nz*being the total number of grid points in the

*x, y,*and

*z*directions, with the terms 0.5[. . .

*] being identical to terms being multiplied by 0.5, except these terms are at the previous time level, and the hat indicates variables that have been interpolated to a cell face. Each equation has been assigned a nonlinear residual, for example,*

^{n}*F*

^{n + 1/2}

_{uρ}, with the equation

**F**(

**x**) = 0 containing all of these nonlinear residuals being solved by the JFNK solution procedure described in the next section.

Interpolated cell-face quantities are computed by the quadratic upstream interpolation for convective kinematics (QUICK; Leonard and Drummond 1995) advection scheme, with these quantities being multiplied by the appropriate normalized cell-face velocity component, *u*_{i±1/2,j,k} = 0.5(*u**ρ*_{i±1,j,k}/*ρ*_{i±1,j,k} + *u**ρ*_{i,j,k}/*ρ*_{i,j,k})Δ*t*/Δ*x*, *υ*_{i,j±1/2,k} = 0.5(*υ**ρ*_{i,j±1,k}/*ρ*_{i,j±1,k} + *υ**ρ*_{i,j,k}/*ρ*_{i,j,k})Δ*t*/Δ*y*, and *w*_{i,j,k±1/2} = 0.5(*w**ρ*_{i,j,k±1}/*ρ*_{i,j,k±1} + *w**ρ*_{i,j,k}/*ρ*_{i,j,k})Δ*t*/Δ*z*. As indicated, all turbulent fluxes are computed on a cell face, for example, *F*^{n+1}_{θxi+1/2} = 0.5(*ρ*^{n+1}_{i} + *ρ*^{n+1}_{i+1})*κ*(*θ*^{n+1}_{i+1} − *θ*^{n+1}_{i}). Note that both advective and turbulent fluxes near a lateral boundary contain a constant in-time environmental component. At the sidewall boundaries of the domain, the velocity normal to the boundary was specified, whereas the velocities normal to the top and bottom boundaries were set to zero. The falling of rain term along with the surface fluxes of energy and gas were approximated by a first-order-in-space backward finite-difference approximation. The surface fluxes were computed based upon a known sea surface temperature and a spatially interpolated surface density field, with these fields being used to define the surface pressure, potential temperature, and saturated mixing ratio. Also, at the sidewall boundaries, the pressure gradient terms utilized a first-order-in-space forward or backward finite-difference approximation.

## 3. JFNK solution procedure

**F**= [

*F*,

_{uρ}*F*,

_{υρ}*F*,

_{wρ}*F*,

_{θρ}*F*,

_{qυρ}*F*,

_{qcρ}*F*]

_{ρ}^{T}, and

**x**is a vector representing the discrete variables [

*uρ*,

*υρ*,

*wρ*,

*θρ*,

*q*,

_{υ}ρ*q*,

_{c}ρ*ρ*]

^{T}. Newton’s method solves Eq. (20) by a sequence of steps defined bywhere

*J*is the Jacobian matrix,andwhere

*i*∈ [1,

*n*

_{max}] and

*j*∈ [1,

*n*

_{max}], with

*n*

_{max}=

*n*

_{tot}×

*n*

_{υar},with

*n*

_{ntot}=

*nx*×

*ny*×

*nz*and

*n*

_{υar}being either 6 (moist bubble) or 7 (hurricane). The iteration over

*k*is continued untilwhere

*ϵ*

_{nl}= 1 × 10

^{−4}is the nonlinear convergence criteria, and the norm in Eq. (24) is defined by ||

**F**(

**x**)||

^{k}_{2}= [Σ

^{nmax}

_{i}

*F*

^{2}

_{i}]

^{1/2}. Equation (21) is solved by the iterative Krylov solver GMRES (Saad 1996) with a right preconditioning option. For each iteration within GMRES the action of the Jacobian matrix on a vector is required. A key property of the JFNK solution procedure is that individual elements of the Jacobian matrix are never formed; instead the action of the Jacobian matrix times a vector is approximated bywherewith

*δ*

**z**a vector used within GMRES and

*a*= 1 × 10

^{−8}. The specifics concerning how the physics-based preconditioner is embedded within GMRES can be found in appendix A.

## 4. Moist bubble

The JFNK machinery supported by the physics-based preconditioner will be used to simulate the buoyant rise of a two-dimensional precipitating moist bubble. The two primary goals of this section are to demonstrate the ability of the JFNK approach to achieve second-order-in-time accuracy on the solution of the nonlinear physical model and to quantify the amount of CPU time required by two different semi-implicit approaches to produce the same level of error as a second-order-in-time JFNK model. As discussed in the introduction, the time-split approaches to be used in the moist-bubble simulations are either the first-order semi-implicit method used in preconditioning the implicitly balanced model or the leapfrog semi-implicit model. Details concerning the leapfrog semi-implicit model can be found in appendix B.

*y*direction, setting all terms associated with the advection and diffusion of scalar quantities in the

*y*direction to zero, neglecting surface fluxes, and using the following discretized form for the energy equation:with the hat quantities being interpolated to the cell center using the QUICK algorithm. Note that the above spatial discretization is the same as used in the leapfrog semi-implicit model.

*t*= 0.0125 s that includes a smooth source function of the formin equations that involve

*ρq*and

_{υ}*ρ,*where

*f*

_{time}= (

*t*− 30 s)/10 s and

*f*

_{space}= dis(

*x*,

*z*)/200 m, with dis(

*x, z*) the distance in meters from the center of the bubble (

*x*= 1290 m,

*z*= 2400 m).

*θ*and

*ρ*set to 300 K and 1 kg m

^{−3}, respectively. The initial

*q*field is set to produce a relative humidity of 50% and the initial

_{υ}*q*field is set to 1 × 10

_{c}^{−6}. The chosen measure of error is the

*l*

_{2}norm, which is defined aswith the potential temperature,

*θ*

_{base}, coming from an implicitly balanced simulation utilizing a time step of 0.0125 s, and the potential temperature,

*θ*, is the computed value.

_{c}*t*. This time step was about one-half the dynamical time scale of the problem,with this time scale being primarily limited by the movement of two well-defined pockets of

_{s}*q*found within the computational domain. The time step for the semi-implicit simulations was limited by the advective time scale of the problem,with the first-order semi-implicit method being able to run up to 0.5Δ

_{c}*t*

_{advec}= 0.5 s and the leapfrog semi-implicit method using a maximum time step that was half this value. Simulations were run for a time period of 600 s with the eddy diffusivity being set to a constant value of

*κ*= 50 m

^{2}s

^{−1}.

Figure 1 shows the potential temperature field from the base solution at the end of the simulation. At this time, both a warm pocket and a cold pocket in the potential temperature field are present. The warm pocket is the result of latent heat release due to the excess of *q _{υ}* produced by the forcing function. The cold pocket is produced by precipitation falling through an unsaturated atmosphere and subsequently evaporating. Of these two regions, the largest region of numerical error, found by comparing simulations against the base simulation, is found in the vicinity of the warm pocket.

Figure 2 reveals difference fields in potential temperature from the various leapfrog semi-implicit simulations and an implicitly balanced simulation in the vicinity of the warm pocket. As evident in the figure, as the time step for the leapfrog semi-implicit model decreases, the difference fields start to approach what was produced by the implicitly balanced approach using a much bigger time step. Thus, the traditional semi-implicit leapfrog approach appears to be significantly less accurate than the implicitly balanced approach for a given time step size.

Figure 3a shows a time step convergence study for the moist-bubble test problem and quantifies results shown in Fig. 2. As evident in Fig. 3a, both semi-implicit algorithms display a first-order-in-time slope, whereas the implicitly balanced algorithm displays a second-order-in-time slope. As quantified in Fig. 3a, the leapfrog semi-implicit procedure is slightly more accurate than the first-order semi-implicit approach, but is still significantly less accurate than the implicitly balanced approach for the same Δ*t*. Since the leapfrog semi-implicit method was not significantly more accurate than the first-order semi-implicit approach, for the subsequent hurricane simulations only results from the first-order semi-implicit approach will be presented. Note that, as discussed in appendix B, the primary reasons for why the leapfrog semi-implicit method did not produce a second-order-in-time slope are the use of an Asselin filter for stability and the computing of diffusion terms at the old time level.

As mentioned in the introduction, a key component of this paper is to quantify how fast a given algorithm can compute a solution to a specified level of accuracy. Figure 3b reveals lines of temporal error versus CPU time required to complete a given simulation from the different temporal integration methods. From the figure it is clear that for the same amount of CPU time the implicitly balanced approach is more accurate than the semi-implicit approaches. Likewise, for the same accuracy, the implicitly balanced approach is more efficient than the semi-implicit approaches. The primary reason for why the implicitly balanced approach is able to produce a given error more efficiently than the semi-implicit approaches can be directly attributable to the ability of the implicitly balanced approach to achieve second-order-in-time accuracy and the physics-based preconditioner to reduce CPU timings by a factor of at least 10. Results from the hurricane simulations presented next will also illustrate the robustness of the implicitly balanced approach.

## 5. Hurricane simulations

As discussed in Skamarock et al. (1997) and Thomas et al. (2003) the inversion of only one implicit equation per time step in multidimensions can be difficult. Now imagine that instead of a single equation, seven coupled three-dimensional equations are being inverted, with this inversion process occurring several times per time step, for example, for each Newton iteration. It may be difficult to envision that the seven-equation system can achieve a given temporal error more efficiently than a one-equation system; however, as will be shown in this section and similar to what has been previously shown for the moist bubble, the implicitly balanced hurricane model will indeed be able to accomplish this feat.

The hurricane model employed for this study is based on the work of Tripoli (1992). Mean winds are assumed to be zero, with potential temperature and *q _{υ}* fields being obtained from the sounding described in Tripoli (1992). Initial velocity fields are obtained by a modified Rankine vortex (see appendix B of Tripoli 1992), with key parameters for the vortex being defined by its relative vorticity

*ζ*= 2 × 10

^{−4}and the radius (500 km) at which the total area-weighted negative vorticity balances the positive vorticity of the prescribed vortex. The domain used in the hurricane simulations is 1500 km × 1500 km × 15 km employing 150 × 150 × 50 grid points, and since the speed of sound is approximately 300 m s

^{−1}, this implies Δ

*t*≈ 1 s. The latitude of the simulations was specified at 20°N latitude, and unless otherwise indicated, the sea surface temperature (SST) was held at a constant value of 302 K. As in the moist-bubble calculations, a constant eddy diffusivity was chosen; however, unlike the previous calculations the eddy diffusivity did vary in the vertical dimension from a value of 30 m

_{s}^{2}s

^{−1}at the surface, linearly increasing with height to a value of 750 m

^{2}s

^{−1}at 3000 m, and above this height the eddy diffusivity was fixed at 750 m

^{2}s

^{−1}. The simulations were run on the parallel Compaq Q machine at Los Alamos National Laboratory utilizing 36 processors with the model domain being broken up into horizontal subdomains and communicating to each other by standard message passing interface software.

*t*= 60 s run for a time period of 3 days. The 3-day integration time period was required to establish a hurricane that was in approximate balance with the environmental forcing. Figure 4 shows isosurfaces of

*q*at the end of this 3-day simulation and reveals a cloud pattern indicative of a hurricane with a distinct cloud-free eye being present in the middle of the

_{c}*q*field. Figure 5 reveals the temporal evolution of minimum surface pressure over this 3-day time period and illustrates that after a rapid pressure drop during the first day of the simulation, the minimum surface pressure slowly decreases to a value indicative of a very intense hurricane.

_{c}*t*≈ 100 s, defined by Eq. (30), with minimum values of Δ

_{d}*t*being primarily produced by advection of

_{d}*ρq*near the eye. Because the simulations were being run for a time period of a day and the computing resources necessary to compute a base simulation for Δ

_{c}*t*< 10Δ

*t*are considerable for a time period of this length, a time step size of Δ

_{s}*t*= 10 s was chosen for the base simulation.

Figure 6 reveals error from the various hurricane simulations as a function of time step size and CPU time. The results shown on Fig. 6 are similar to what was found for the moist bubble, with the implicitly balanced model once again producing a given error more efficiently than the first-order semi-implicit model. But, unlike the moist-bubble simulations, a loss of first-order-in-time accuracy occurs for the larger time step sizes of the first-order semi-implicit model, and this may be the result of the model exceeding the advective time scale of the problem and/or oscillations produced by calculating terms such as diffusion at the old time level.

Figure 7 reveals that after starting off with a zero error for all simulations, for the first-order semi-implicit simulations for Δ*t* > Δ*t _{s}* the error grows rapidly in time and becomes much larger in magnitude than the first-order semi-implicit simulation with Δ

*t*= Δ

*t*and the error from the implicitly balanced simulation. In fact, for the first-order semi-implicit simulation with Δ

_{s}*t*≈ Δ

*t*, the error, in agreement with Fig. 6, begins to just approach the error from the implicitly balanced simulation with Δ

_{s}*t*= 60 s. Figure 8 further demonstrates that differences between the base simulation and the first-order semi-implicit simulations decrease notably as the time step nears Δ

*t*, with the primary differences being found in the vicinity of the eye.

_{s}The ability of the physics-based preconditioner to significantly reduce both the average number of Krylov iterations taken per time step and the CPU time for the implicitly balanced model is shown in Table 2. In fact, with the physics-based preconditioner active the implicitly balanced model is able to run faster than real time for time step sizes between 40 and 60 s.

## 6. Conclusions and future work

A three-dimensional implicitly balanced numerical model designed to simulate deep moist convective systems such as hurricanes has been described in this paper. This is the first application of the JFNK approach to a highly nonlinear three-dimensional system that is representative of a convective process that routinely occurs in nature. For the problems shown in this paper for which a large separation in time scales was present, the second-order-in-time implicitly balanced model was able to produce a given error using much larger time steps than from semi-implicit approaches using much smaller time steps. For the same time step size, the error from the implicitly balanced approach can be several orders in magnitude smaller than that from a semi-implicit model. Additionally it was shown that the JFNK method can produce this small error in an efficient manner because of the physics-based preconditioner ability to reduce by a factor of 10 the number of Krylov iterations.

Because both semi-implicit approaches are Eulerian and are limited by the advective time scale, the maximum allowable time step is smaller than what could be used by a semi-implicit semi-Lagrangian approach (Benoit et al. 1997). Future work should compare the efficiency and accuracy of a semi-implicit semi-Lagrangian approach against the implicitly balanced approach advocated in this paper. The comparison should include flow regimes for which the dynamical time scale of the problem is limited by advective processes.

In this paper a second-order-in-time Crank–Nicolson time-differencing formulation was employed; however, with minor modifications third-order-in-time semi-implicit or implicit Runge–Kutta formulations could be readily employed in the implicitly balanced framework and compared against the Crank–Nicolson approach. For example, though a third-order-in-time semi-implicit Runge–Kutta approach involves two implicitly balanced solves per time step, as illustrated in this paper, the cost of this additional nonlinear solve may be offset by the much lower error produced by the higher-order time-stepping algorithm. Upon applying the Runge–Kutta approaches to the simulation of historical hurricanes, it would then be interesting to quantify whether or not these higher-order-in-time approaches have an appreciable impact on the ability of a numerical model to accurately forecast movement and intensification of a hurricane over a relatively long integration time period. Thus, a key question that could be addressed from the analysis is as follows: What is the minimum level of temporal accuracy required to reproduce a historical hurricane for which environmental conditions were changing rather rapidly during the lifetime of the hurricane?

Another important aspect of the implicitly balanced approach is that physical models be relatively smooth to ensure convergence of Newton’s method. For example, the standard bulk approach used to convert cloud water into rainwater is discontinuous, with Lopez (2003), showing that this approximation should be modified to ensure the success of an adjoint model. The development of smooth physical models is a time-consuming task, but this process needs to be done before progress can be made with respect to developing a truly predictive model. Also, it is important to realize that even when a parameterization is smooth, the parameterization can induce changes in time of mean variables that are even faster than those created by sound waves. Hence, for efficiency, parameterizations with fast time scales should be implicitly treated within the physics-based preconditioner.

This work was supported by the Laboratory Directed Research and Development Program of Los Alamos National Laboratory, which is under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy under DOE Contracts W-7405-ENG-36 and LA-UR-03–9158. Computer resources were provided both by the Computing Division at Los Alamos and a small Linux cluster maintained by Dave Moulton in the Theoretical Division at Los Alamos. Thanks also goes out to Jim Kao for the code used in calculating the initial conditions for the hurricane simulations and for constructive comments made by anonymous reviewers.

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# APPENDIX A

## Details of Physics-Based Preconditioner

This appendix provides details concerning how a preconditioner is embedded within GMRES as well as specifics concerning the physics-based preconditioner used in the hurricane and moist-bubble simulations.

*δ*

**x**

*is computed in an iterative manner (m is the Krylov iteration):and the vector*

^{k}**v**

_{gmres}and the coefficients

*η*are computed by the GMRES algorithm. Equation (A5) is iterated untilwherewith

*γ*= 10

^{−2}. Typically, the value of

*ϵ*varies from a relatively large tolerance for the first Newton iteration to a rather tight linear tolerance for the last Newton iteration. Hence, this process reduces the amount of work when the nonlinear residual is large and increases it when linear accuracy helps the convergence of the nonlinear iteration.

_{L}**v**

_{gmres}, and computes the vector

*P*

^{−1}

**v**

_{gmresw}. The resulting vector is then used in a matrix-free approximation, that is, Eq. (25), of the Jacobian matrix times the vector. Note, that these two steps make up the primary computational piece of the implicit solver, with any gains in efficiency coming from either increasing the efficiency of the physics-based preconditioner itself and/or increasing the ability of the physics-based preconditioner to reduce the number of GMRES iterations.

**x**and the next time step of the time-split semi-implicit process begins. Obviously this process is computationally more efficient than the implicitly balanced solver that requires the preconditioning process to be computed for each GMRES iteration; however, as documented in this paper and in Reisner et al. (2003) and Mousseau et al. (2002) the increase in efficiency is offset by the decrease in accuracy of the semi-implicit approach.

*q*and

_{υ}*q*, and the falling of precipitation explicit in time, the temporal discretizations in Eqs. (12)–(19)—which are now employed in the semi-implicit physics-based preconditioner—are expressed in the momentum equations as follows:The energy equation used in the physics-based preconditioner for the hurricane simulations is expressed as follows:whereas the energy equation used in the physics-based preconditioner for the moist-bubble simulations is expressed as follows:The conservation equations and the equation of state used in the physics-based preconditioner are expressed as follows:whereTo rewrite the discretized equations in a form similar to Eq. (21), the following expressions are required:where the asterisk indicates the current state of the linear solution. The expressions Eqs. (A22)–(A28) are substituted to obtain the following:The delta form of the energy equation for the hurricane simulations is the following:whereas the delta form of the energy equation for the moist-bubble simulations is the following:The delta form of the conservation equations and the equation of state can be expressed as follows:wherewith the right-hand sides of Eqs. (A29)–(A36) coming from the vector,

_{c}**v**

_{gmres}.

*δuρ*,

*δυρ*, and

*δwρ*in Eqs. (A29)–(A31), the individual velocity components are substituted into the energy equation, (A32), to obtain an equation of the formwith the coefficients (

*a − m*) being computed numerically using a procedure described in Reisner et al. (2001). The variable rhs

*is obtained by initially assuming that*

^{si}*p*= 0 in Eqs. (A29)–(A31), forming and then substituting the resulting cell-face velocities into Eq. (A32), and after executing the advection scheme, combining the resulting term with

*υ*

_{gmresρθ}to obtain rhs

^{si}

_{i,j,k}.

This completes the description of the necessary components of the physics-based preconditioner. When employing the semi-implicit method as a preconditioner to Eq. (A9), the following steps are conducted in each call to the preconditioner:

- If a new time step, then compute the coefficients of Eq. (A39) from old time values of
**x**. - Given
**v**_{gmres}construct the*rhs*^{si}_{i,j}in Eq. (A39). - Approximately solve Eq. (A39) for
*δρθ*. - Given
*δρθ*, compute*δ***u***ρ*,*δ***v***ρ*, and*δ***w***ρ*from Eqs. (A29)–(A31). - Use the newly computed velocities to solve for
*δρ***q**in the following equation:_{v} - Use the newly formed
*δρ***q**to update_{v}*f̂*_{cloud}and then solve for*δρ***q**and_{c}*δρ*. - Pass the newly updated vector,
*δ***z**, containing*δ***u***ρ*,*δ***v***ρ*,*δ***w***ρ*,*δρθ*,*δ***q**_{v}*ρ*,*δ***q**_{c}*ρ*, and*δρ*to the Krylov solver.

For the moist-bubble simulations a relatively simple v-cycle multigrid solver employing symmetric successive overrelaxation (SSOR) as the smoother was chosen to invert the semi-implicit matrix (A39). When used as a solver, a linear tolerance of 1 × 10^{−6} was specified; however, when the first-order semi-implicit system was used as a preconditioner for the implicitly balanced system, a fixed number of v cycles and SSOR iterations was used such that the linear residual was significantly reduced over its initial value.

For the hurricane simulations, GMRES along with an alternating-direction implicit preconditioner (ADI; Skamarock et al. 1997) was used to solve the semi-implicit matrix. Implicitly balanced hurricane simulations employed three GMRES iterations with one ADI iteration per GMRES iteration, whereas the number of GMRES iterations varied in the first-order semi-implicit simulations such as to produce a linear tolerance of 1 × 10^{−6}.

# APPENDIX B

## Details of Leapfrog-Based Semi-Implicit Scheme

This appendix provides details concerning the leapfrog-based semi-implicit approach used in the moist-bubble calculations. For stability and efficiency the nonlinear diffusion terms are calculated at time level *n* − 1 (Wicker and Skamarock 2002); however, implicit time-differencing schemes for the diffusion terms could be used to improve the overall temporal accuracy of the approach (Ascher et al. 1995). The formulation of the semi-implicit method is based on work by Kwizak and Robert (1971) and Robert et al. (1972). Note that the current algorithm will produce a second-order-in-time slope for a dry bubble (Reisner et al. 2003) under the following assumptions: a linear equation of state, no diffusion, and for a relatively short integration time period. For longer times, the use of an Asselin–Robert time filter is required for stability, and as noted by Durran (1991), the use of an Asselin–Robert time filter degrades the temporal accuracy of the leapfrog time differencing to first order.

*p*

^{n+1}=

*p*

^{n−1}+ ∂

*p*/∂

*ρ*

*θ*|

^{n−1}2Δ

*ρ*

*θ*, with Δ

*ρθ*=

*ρθ*

^{n}^{+1}−

*ρθ*

^{n−}^{1}in the leapfrog formulation. In Eq. (B3) the advection of

*ρθ*has been broken up into an advective part and a divergence part, with the divergence part being further broken up into two parts, with the second part being centered at time level

*n*. The primary difference between the leapfrog semi-implicit discretization and the first-order semi-implicit discretization is that all terms, except for diffusion, are time centered in the leapfrog approach.

*δuρ*=

*uρ*

^{n}^{+1}−

*uρ*

^{n−}^{1}, into Eqs. (B1)–(B7) to obtain equations in delta form (not shown). Next, the delta form of the momentum equations are substituted into the delta form of the energy equation to obtain an equation of the formBecause the cloud physics terms are time centered, unlike in the first-order semi-implicit model, the terms do not directly appear in the semi-implicit leapfrog model and are found within rhs

^{leap}

*. Hence these terms do not need to be evaluated for each iteration of the leapfrog semi-implicit solution procedure; however, differences in CPU timings between the two approaches were relatively small, with no serious attempt being made to optimize either numerical procedure. Once Eq. (B8) has been solved to a linear tolerance of 1 × 10*

^{si}^{−6}using a simple v-cycle multigrid solver, then all terms in Eqs. (B1)–(B7) can be updated. After this step has been completed, application of the Asselin filter is conducted, which then completes the leapfrog semi-implicit update procedure.

Values of constants used in the physical model.

Average number of GMRES iterations per time step and CPU time in seconds required to complete a 720-s integration period. Values were computed at the end of implicitly balanced hurricane simulations with (Pre) and without (NoPre) physics-based preconditioning.