a. Mass variables

The atmosphere we consider is made of dry air and water substance. The water substance is recognized in two phases, vapor and condensate (liquid or solid). The condensate is further classified in two modes, airborne and precipitating. The distinction occurs from the difference in implied particle size. Due to its small particle size, airborne condensate (cloud) is in instantaneous equilibrium and moves together with the environment of moist air in which it is suspended. Due to its greater drop size and smaller surface-to-volume ratio, the precipitating condensate (rain or snow) needs time to adjust itself to the environment through which it falls. Thus, the spatial distribution of precipitating water must be predicted separately from that of airborne moisture (the sum of vapor and airborne condensate), with due consideration of the conversion between phases and between modes.

The estimation of those conversion rates is greatly facilitated by the adoption of a Kessler-type parameterization of the microphysics. The boldest assumption by Kessler (1969) is to collapse the drop size–dependent processes into a set of a few formulas written in terms of the precipitation water content (a single scalar), by integrating them over an assumed drop size distribution. The specific formulas, adopted from Klemp and Wilhelmson (1978), are listed in appendix A.

Ideally, the liquid and ice phases of the condensate should be separate. In active clouds, however, the two phases are hardly in equilibrium; and, moreover, the ice has various crystal forms and various degrees of aggregation. In order to avoid the complication of a nonequilibrium mixture, O90 synthesized a single, continuous phase of condensate that would behave, with respect to vapor, like liquid or ice, depending on temperature. The synthesis was made by smoothing the Kirchhoff equation across a freezing zone of temperature, in such a way that all three phases of water were consistently modified. The smoothed saturation vapor pressure was practically unchanged from the original, either over liquid or over ice depending on temperature, but the latent heat of fusion was replaced by a large anomaly of the specific heat of the synthesized condensate in the freezing zone. This approximation is adopted in the present model. Accordingly, we do not introduce additional formulas for parameterizing ice–form interactions, except for reducing the fall speed of our single-phase precipitation when the temperature is below the freezing point.

The mass variables are expressed as densities in geometrical space. Since the symbols used in O90 for this purpose have proved to be rather unpopular, we revert here to the more traditional density symbol, ρ, with subscripts. Namely, ρ_a stands for dry air, ρ_υ for vapor, ρ_c for airborne condensate (cloud), and ρ_r for precipitating condensate (“rain” and “snow” in our extended sense). We also define ρ_m = ρ_υ + ρ_c for the total airborne moisture, and ρ = ρ_a + ρ_m + ρ_r for the total mass per unit volume. As discussed below, ρ_υ and ρ_c are not independently predicted, but diagnostically separated from the predicted ρ_m.

b. Entropy

The first law of thermodynamics concerns energy conservation and can be expressed in terms of the internal energy, entropy, or enthalpy. In a model of the moist atmosphere on the geometrical coordinates, the entropy, in our view, is the most convenient variable to express the conservation law. In the meteorology of the dry atmosphere, the potential temperature is the prevalent variable employed for the same purpose, but it is merely an exponential transform of the entropy expressed in units of temperature. The equivalent potential temperature is also widely used in studies of the moist atmosphere, although it is only approximately conservative due to the assumption of the pseudoadiabatic process. Various proposals, such as Betts (1973) and Tripoli and Cotton (1981), have been made to define a potential temperature that more rigorously handles the phase changes of water substance. Hauf and Höller (1987) have argued that those definitions are covered by the entropy temperature, which they defined directly from the entropy. In the present paper, the entropy itself is used as one of the prognostic variables. If so desired, a conversion of the entropy to an equivalent variable in temperature units can be easily made from the model output.

If all the processes are adiabatic and reversible, the entropy is strictly conserved. This is assumed to be the case with respect to phase transitions of water substance within the moist atmosphere. On the other hand, certain important processes in the real atmosphere, such as the diffusive mixing or the generation of precipitation, are irreversible. The common practice in modeling is to add the effects of those irreversible processes to the conservation equations in the form of parameterized sources and sinks. In this regard, the present model shares the same basic premise with other dynamic models of the atmosphere. Nevertheless, it will be prudent to ascertain that the employed parameterization does not violate the second law of thermodynamics. Further discussions on the second law are given in appendix B.

One unique feature in the formulation of the present model is our treatment of the pressure. The prevailing practice in atmospheric modeling is to predict the pressure by a tendency equation which is derived from the combination of dynamic and thermodynamic principles. Although the derivation for the dry atmosphere is straightforward, the corresponding one for the moist atmosphere becomes entangled with the complexity of moist thermodynamics. According to the classical dictum of thermodynamics, the pressure is a state variable that, along with the temperature, can be diagnostically determined from the mass and entropy. Thus, it was proposed by O90 to separate the dynamics out of the determination of the pressure. Even in hydrostatic models, as discussed by O90 and demonstrated by DeMaria (1995), this approach eliminates Richardson's problem of an awkward “w equation” on the height-coordinate. In nonhydrostatic moist models, the same approach also allows a stylistically simple yet accurate set of prognostic equations and facilitates the logistics of solution.

The entropy per unit volume may be defined for each mass component in the volume, that is, σ_a for the dry air, σ_m for the airborne moisture (vapor and cloud), and σ_r for the precipitating water. In the present model, however, only the total entropy density, σ = σ_a + σ_m + σ_r, needs to be predicted. In particular, as explained below, it does not require a separate prediction of σ_r.

c. Temperature

In the absence of precipitation, O90 defined two equilibrium states of moist air that were possible for a set of predicted values of ρ_a, ρ_m, and σ. State 1, at temperature T₁, is characterized by the absence of condensate, even if the vapor is supersaturated. State 2 is characterized by the vapor being saturated at temperature T₂, even by evaporating borrowed water if necessary. The two temperatures are diagnostically determined by solving, respectively,

View Expanded

for known thermodynamic functions S₁ and S₂ (see appendix B). For reasons discussed in O90, the realistic temperature T, without supersaturation or borrowed water, should be the greater of T₁ and T₂, while T₂ is always identifiable as the wet-bulb temperature.

It is noted that (2.2) is based on an idealization that the temperature of raindrops is at the wet-bulb temperature in unsaturated air. Kinzer and Gunn (1951) observed the relaxation time of raindrop temperature to be approximately 4.35 s. This rapid relaxation is due to the small amount of heat required to adjust the temperature of a raindrop as compared to that required to evaporate it. Thus, in unsaturated air, ρ_r must be predicted with a parameterized evaporation rate, but σ_r is a known quantity once the wet-bulb temperature is diagnosed. The exchange of both the sensible and latent heat between the raindrops and gaseous environment is internal to the prediction of the total entropy. In saturated conditions, the temperature itself is the wet-bulb temperature, and there is thermodynamically no difference between cloud and precipitation.

d. Pressure and other state variables

Once the temperature is known, the total pressure, p, is given by

pp_ap_υ(2.5)

where the partial pressures, p_a for dry air and p_υ for vapor, are determined by

View Expanded

where E(T) is the saturation vapor pressure. The vapor density and cloud water content are

View Expanded

Note that, if T = T₁, ρ_m = ρ_υ and ρ_c = 0.

a. Definitions and clarifications

Our working model is two-dimensional in the vertical plane and can be run in either of two modes: slab symmetric or axisymmetric. Since the model will eventually be three-dimensional, the dynamic equations below are given in 3D on an f plane, written in Cartesian coordinates (x, y, z). We use vector notation for the horizontal (x, y) components, and the vertical component will be written separately. This is because the fall speed of precipitation W is purely vertical, and its place in the vertical momentum equation needs to be highlighted. Thus, (v, w) denotes the velocity of both dry air and airborne moisture (vapor and cloud), while (v, w_r) denotes the velocity of precipitating water, with w_r = w + W. The sign convention for w and W are the same, so that W is always negative (downward).

When ρ_c (cloud) is present in saturated air, it may be converted to ρ_r (precipitation), by autoconversion and collection processes, at a finite rate Q_r. When ρ_r is present in unsaturated air, it evaporates back to ρ_υ (vapor) at a finite rate, which will be denoted by a negative value of Q_r. Since both ρ_υ and ρ_c are part of the total airborne water ρ_m, we may treat Q_r, either positive or negative, as the conversion rate between ρ_m and ρ_r. Parametric formulas for Q_r and W are given in the appendix A.

Other physical processes, such as radiative heat transfer, boundary layer eddy fluxes, and internal diffusive processes, are not included in the present discussion. A Newtonian cooling may be added to the entropy prediction, but a more realistic formulation of radiative fluxes is outside the scope of this paper. Model experiments that run more than a few hours should include interactions of the atmosphere with the underlying surface. Although a conventional formulation of the constant-flux surface layer and a crude formulation of eddy fluxes in the mixed layer are included in our model, these are not utilized in the experiments reported in this paper.

The omission of internal diffusion terms from our equations is deliberate and requires a clarification. In general, models of finite resolution must avoid the accumulation of spectral power near the Nyquist wavelength (2Δx). Such a problem may occur due to either the inherent spectral cascades in nonlinear dynamics or the numerical accuracy in handling the smallest scales. Although these causes are often inseparable, the problem is usually abated by the inclusion of diffusion terms, attributed to “subgrid-scale” turbulence. In the present model, a low-pass filter that is built into the spectral transform (see section 5b) is applied to every prognostic variable at each time step. Although the filter is designed primarily for reducing the spectral representation error in 2 to 3Δx waves, it also effectively eliminates the spectral accumulation in these scales that is due to nonlinear cascades. Thus, additional diffusion terms are found unnecessary for computational stability and are omitted in our equations. On the other hand, since the built-in filter has a sharp cutoff taper of the sixth order, the filter will not override the added effects of the second- or fourth-order diffusion, if such effects are called for by the model physics. (See appendix C for a demonstration.)

b. Conservation equations for mass and entropy

The continuity equations for the mass variables, ρ_a, ρ_m, and ρ_r, are given by

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These three equations are independent and may be combined into other convenient forms. In particular, the sum of the three gives the equation for the total mass,

_tρρv_zρwρ_rW(3.4)

Obviously, there is no internal source of mass; but, since W does not vanish at the ground, the second term of the vertical mass flux implies a loss of precipitating water to the ground.

c. Conservation equations for momentum

The horizontal momentum equation for the total mass, including precipitation, is

View Expanded

where, in the third term, ρ_rW supplements ρw to account for the vertical flux of horizontal momentum by precipitation.

Since the pressure gradient force acts on volume, it does not directly accelerate the condensate whose volume is negligibly small. Nevertheless, even in accelerating conditions, no relative horizontal motion is assumed between the condensate and gaseous environment. This implies that a frictional drag force instantaneously eliminates any velocity difference. For the total mass, the action and reaction balance out internally.

The circumstance is a little different in the vertical. To illustrate the problem, we write the vertical momentum equations separately for ρ_am = ρ_a + ρ_m and ρ_r. The vertical component of velocity is w for the former, and w_r = w + W for the latter. Specifically,

View Expanded

where F_r is the vertical component of frictional drag force acting on ρ_r, and its reaction on ρ_am. The pressure gradient force on ρ_r is, again, negligible.

The sum of (3.7) and (3.8) yields an equation for the total vertical momentum:

View Expanded

in which F_r is balanced out, but a source term, Q_rW, remains as the residual of −Q_rw and Q_rw_r.

To explain the cause of this residual source, let us assume for the moment that Q_r is positive. Then, the term −Q_rw on the rhs of (3.7) represents a rate of momentum loss in ρ_am simply due to the fact that part of ρ_m moving with velocity w is reclassified to be part of ρ_r. The same reclassified mass appears in (3.8) as a momentum source but with velocity w_r. This sudden acceleration is due to the gravity acting on raindrops (or snow) before it becomes fully opposed by F_r. Therefore, the source term is legitimate, except for the idealization that the acceleration takes place instantaneously as soon as the parameterization reclassifies airborne condensate to precipitation. If Q_r is negative, it is an instantaneous deceleration that accompanies the reclassification of precipitation to vapor.

As a prognostic equation, however, (3.9) has a logical contradiction. The contradiction is actually inherited from (3.8), which considers w_r to be a prognostic variable. Since w itself is predictable by (3.7), these equations are in fact predicting W, in spite of our assumption to parameterize. In terms of prognostic calculations, (3.9) also presents practical difficulties to a numerical scheme which requires a priori specification of the lower boundary condition on (ρw + ρ_rW). An attempt to ignore only the local time derivative of W leaves physically unacceptable source/sink terms in the kinetic energy budget. Theoretically, we can choose to predict W, by removing it from the list of parameterized variables. This option, however, may lead to the total abandonment of the microphysics parameterization, because W is intimately involved in the estimation of F_r and Q_r. In order to stay the course, (3.8) and (3.9) must be somehow modified to be compatible with the diagnostic determination of W.

This dilemma has a well-known precedent in the quasistatic system of equations. The customary resolution is to ignore the material derivative of w in the equation of vertical motion, while w is diagnostically derived from the hydrostatic equation (e.g., O90), although this leads to a counterintuitive consequence that the kinetic energy budget will not contain the contribution of the vertical motion. As observed by Eliassen and Kleinschmidt (1957, 20–21), this is commonly accepted as a plausible approximation without rigorous justification.

In the present nonhydrostatic case in which w is fully prognostic, an analogous principle may be applied to the terminal fall speed W. Namely, we postulate that the material derivative of W along the path of precipitation,

D^(r)_tW_tWvWw_rzW,(3.10)

be ignored in (3.8), which then is reduced to

View Expanded

Similarly, (3.9) for the total vertical momentum is replaced by

View Expanded

which is the sum of (3.11) and (3.7).

Although the kinetic energy equation is not used in prediction, it is interesting to see how it comes out of the new set of prognostic equations. From (3.6) and (3.12) with the aid of (3.1)–(3.4), an equation for the total mechanical energy is derived as

View Expanded

with the definitions

View Expanded

where K and Φ are the total kinetic and potential energy per unit volume, respectively; and K_r and Φ_r of the precipitation alone are defined for the convenience of notation, although these are part of the total. As expected, the kinetic energy does not contain any contribution from W in this energy budget of the modified system.

On the rhs of (3.13), the first term is the mechanical work done by the changing specific volume against pressure and constitutes the link with the internal energy of gaseous matter. The second term is always negative, as W is, and represents the dissipation of kinetic energy of precipitation due to work done by the frictional drag against falling precipitation (i.e., F_rW), although F_r is replaced by the nearly equivalent gρ_r under the present approximation. In the third term on the lhs of (3.13), the extra flux by W does not vanish at the ground and represents a loss of the mechanical energy from the atmosphere; K_r will be lost on impact, though, on elevated terrain, Φ_r is still retained by the ground water so that it can run downhill.

In conclusion, we shall accept (3.12) as a plausible and consistent approximation for the vertical momentum equation. In more practical terms, numerical discrepancies that may arise from ignoring (3.10) are expected to be slight, because W varies rather slowly with a small fractional power of ρ_r and the square root of ρ_a, with possible minor exceptions in the melting zone.

a. The equations in advective form

For the convenience of numerical calculations, the equations in the preceding section are converted to equivalent equations in advective form. Since our numerical method utilizes differentiable spectral bases, the conversion does not alter the conservativeness of the original flux form.

For the mass equations, (3.1) through (3.4) are now written as

View Expanded

Of the last three equations for water substance, only two are independent and actually used in prediction. Our preference is (4.4) and (4.5).

For the horizontal and vertical components of velocity, (3.6) and (3.12) respectively yield

View Expanded

b. Background and deviations

The mass and thermodynamic states are strongly stratified in the atmosphere. By predicting deviations from predefined background states, rather than the full values of variables, we may gain a few decimal places in numerical accuracy. Such a gain is not trivial, since the model repeats prognostic calculations in small time steps a great number of times.

In general, the background may vary in space. At the present, it is defined to vary only in the vertical and will be indicated by a circumflex on appropriate symbols. For example, ρ̂_a, ρ̂_m, σ̂, p̂, etc. are functions of z defining the background states of respective variables. No background is assumed for precipitation, so that ρ̂_r = 0 and ρ̂ = ρ̂_a + ρ̂_m.

Although any part of the definitions for the background states may also be used for defining the initial states of a particular model run, the two states are conceptually independent; an individual run can be started from any set of initial states as long as it is physically possible.

The prognostic equations (4.2)–(4.8) are rewritten in terms of deviations, indicated by a prime. Since the transformation is rather mechanical, only a few examples are listed below.

With a shorthand notation, a, for the log-density of dry air (normalized by a constant reference density ρ₀),

View Expanded

the mass continuity (4.2) becomes

D_taw_zâv_zw(4.11)

and, for the total water substance, (4.5) becomes

D_tμw_zμ̂ρ⁻¹_azρ_rW(4.12)

where, since μ̂_r = 0,

μ̂μ̂_mμμμ̂μ^′_mμ_r(4.13)

Another example of using deviations is given below.

c. Pressure gradient

From the predicted entropy and mass variables, the pressure at any spatial point can be determined by the diagnostic procedures of section 2. However, it is not the pressure itself but its spatial gradient that is required by (4.7) and (4.8). Since advective terms in those equations are calculated by the spectral transform method, it is not advisable to employ a finite-difference method to evaluate the pressure gradient. It was proposed in O90 to calculate the gradient with the aid of analytic differentiation of pressure in the manifold of thermodynamic variables. This method, derived for reversible thermodynamics, does not work well in the present case which includes irreversible processes of precipitation.

The pressure gradient term in (4.7) is now calculated by

ρ⁻¹pμ⁻¹ττa(4.16)

and the same in (4.8), but together with g, by

View Expanded

where (4.9) has been used in the derivation. Note that a is the log-density of dry air and μ the mixing ratio of the total water substance, as previously defined.

It is significant that the rhs of (4.17) avoids a severe numerical degradation that would occur if the two large terms on the lhs were directly calculated to yield a small residual. When precipitation is present, μ_r is the most prominent contributor to μ′, and the last term, gμ′, in (4.17) represents the so-called water loading effect.

The new strategy for evaluating the pressure gradient via spectral transform costs a little more computation time but works extremely well. In fact, we have been able to remove all ad hoc diffusion terms which were once included in the prognostic equations, since no noise occurs that must be suppressed by such means.

d. Hyperbolic transform of mixing ratios

A purely computational problem of negative water can occur in any numerical model of finite resolution. In our spectral model, it occurs in sidelobe oscillations due to Gibbs' phenomena. The sidelobes are not the reflection of spectral aliasing or other inadvertent error, but are simply the result of discarding the unresolvable part of would-be continuous spectra. Their amplitudes, in practice, are found to be less than a few percent of the main lobes; and, if left untouched, they normally incur no cumulative or permanent damage to model runs, except for the case of water substance.

Near the edge of a cloud, especially above the cap of a rising convective cell, the predicted mixing ratio, μ or μ_r, may become negative in the sidelobes and cannot be interpreted thermodynamically. When finite-difference models encounter similar difficulties, some propose to redistribute negative values to neighboring grid points, or some others employ the upstream difference scheme to avoid generation of negative values. The definition of a neighborhood in the former is arbitrary, and the scheme of the latter is very dispersive. There are more elaborate schemes to preserve positiveness (e.g., Shchepetkin and McWilliams 1998), which we are not quite ready to adopt.

Another way of coping with the problem is to predict a transformed variable. For example, if ν = lnμ is predicted, the recovered μ = expν will always be positive, regardless of Gibbs' phenomena in ν. However, the logarithmic function bends μ at all values; and, in practice, the mean of μ is not well preserved. As a better alternative, we propose a functional transform by one branch of a hyperbola, in which μ is stretched only when it is small enough to be meteorologically insignificant.

Let us consider first a hyperbolic transform and its inverse, defined by

View Expanded

where μ₀ is a constant (typically 10⁻⁷), and a numerical factor 0.5 on the first equation eliminates the necessity of coefficients in the second. This transform is practically linear; that is, ν ∼ 0.5μ, for μ ≫ μ₀. If ν becomes negative, μ will remain positive. Unfortunately, this simple transform is invalid at μ = 0; thus, it cannot be applied to the mixing ratio of condensate. Therefore, (4.18) is slightly modified to extend its validity to μ = 0.

The biased hyperbolic transform and its quasi-inverse are written in terms of two functions, bhyp(μ) and ahyp(ν), which are defined by

View Expanded

The transform between μ and ν is again linear for μ ≫ μ₀, and bhyp(μ) is valid at μ = ν = 0. If ν is negative, the strict inverse of (4.19) will find a negative μ but only within a limited range: −μ₀ < μ < 0. Therefore, the maximum adjustment of negative μ by ahyp(ν) does not exceed μ₀.

The proposed adjustment is admittedly pragmatic but its meteorological significance (or insignificance) may be measured in terms of the potential temperature difference, Δθ, between whether the amount of water represented by μ₀ is in vapor or fully condensed. Our typical value, μ₀ = 10⁻⁷, yields Δθ < 3 × 10⁻⁴ K, which we consider to be small enough. We have made a number of test runs in vigorously convective situations. The results showed no discernible difference between μ₀ = 10⁻⁷ and 10⁻⁸, while very slight differences were detected in the case of μ₀ = 10⁻⁶.

The transform by bhyp(μ) is required only for preparing the initial ν before a model run, and there would be no problem in restricting the initial input of μ to be zero or positive. The prediction is made in terms of ν, and it may run freely positive or negative. Although μ at each time step must be recovered by ahyp(ν) as input to thermodynamic diagnosis, ν itself is untouched, so that the effect of μ adjustments does not accumulate in the predicted ν.

Applying (4.19) to the two variables μ = μ_m + μ_r and μ_r, we may rewrite (4.4) and (4.5) as

View Expanded

where ν_r = bhyp(μ_r), ν = bhyp(μ), and

View Expanded

Actual prediction of ν is performed in terms of its deviation ν′ from the background ν̂ = bhyp(μ̂).

a. Conditions for ideal nesting

In order to allow the large-scale environment to interact with convective clouds, the model is configured on horizontally nested domains of varying resolution. In the vertical there is no nesting, but the resolution may be optionally varied with height by the use of a stretched coordinate.

For prediction of a hyperbolic system of equations, nested models have been known for numerical noise that is generated as waves attempt to cross a domain interface. By adopting a nesting strategy, we accept the obvious fact that certain short waves in one domain are geometrically unresolvable in another of a coarser resolution. No numerical scheme can alter this fact. However, it must also be recognized that those would-be-resolvable waves are not necessarily transmissible to the coarse domain. Transmissibility depends on dynamics and is determined by the existence of a matching time-frequency on both sides. It can be improved by accurate numerical methods but never beyond the resolvability. In designing a nesting strategy, therefore, we must observe the following conditions.

1. The gap between resolvable and transmissible waves should be minimized.

2. Nontransmissible waves should not be allowed to reach the interface. If they do, they will be totally reflected, usually as waves in the computational mode with negative group velocities.

3. The computational phase speed of transmissible waves must be identical on both sides of the interface. A mismatch of phase speeds will cause partial reflection.

4. The interface condition for any field variable is simply that its value and spatial derivatives of every order are matched across the interface. Then, provided conditions 2 and 3 are met, the transmissible waves will be oblivious to the presence of the interface.

The harmonic spectral method is ideal in many respects, but it does not accept any boundary condition except for periodicity. There have been proposals (e.g., Tatsumi 1986) for a modified harmonic spectral method which incorporates a few nonperiodic bases to satisfy arbitrary conditions at the exterior boundary of a finite domain, but the idea is not extendable to two-way nesting over multiple domains. A series of Chebyshev functions can also accommodate boundary conditions (Fulton and Schubert 1987), but the nonuniform spatial representation, extremely skewed toward the boundary, makes it a poor choice for atmospheric applications.

Among the gridpoint methods, second-order difference schemes are employed in many nested models, such as Kurihara and Bender (1980), Zhang et al. (1986), Chen (1991), and Grell et al. (1995). However, the dispersion characteristics of second-order schemes are hardly ideal in view of the conditions stated above. If the mesh size is Δx in a fine domain and doubled in an adjacent coarse domain, the threshold wavelength of transmissibility (by the centered scheme with Courant number 0.5) is 11.5Δx; waves shorter than this limit are totally reflected, and longer waves (up to 20Δx, depending on the assumed tolerance level) cause partial reflection. The reflected waves, which are pure noise in the computational mode, can be removed by appropriate filters or damping schemes, but it is more important to narrow the spectral range of nontransmissible waves by employing accurate numerical schemes. This can be achieved by the use of higher-order difference schemes (e.g., Tremback et al. 1987) but necessary studies of interface conditions for nesting have not been undertaken.

b. The SAFER method

The nesting strategy we have adopted is the spectral representation of spatial fields by cubic B-splines, which are assigned one each at equally spaced nodes in a domain and form the basis functions. Any field, represented by a set of amplitudes at the nodes, is continuously differentiable up to the second order, and the third-order derivative is piecewise continuous. The idea was first applied to Global Atmospheric Research Program Atlantic Tropical Experiment data analysis by Ooyama (1987) and further developed as the architectural foundation for domain nesting in prognostic models. The algebraic detail of this numerical method, called the Spectral Application of Finite-Element Representation (SAFER), was described by DeMaria et al. (1992), including the definition of spline-spectral transforms, the low-pass filter built into the transform, the general form of boundary conditions, and the logistics of exchanging data between adjacent domains. While barotropic hurricane track forecasting was the only application discussed by DeMaria et al. the method itself is sufficiently accurate and versatile for application to numerically more demanding problems. The result of a test with a nonlinear flow problem (Straka et al. 1991) is given in appendix C, although this example is in a single domain and does not involve nesting. For the present purpose, a few significant aspects of the method may be noted below.

For wavelengths greater than 3Δx, the phase speed is very accurate provided that the Courant number does not exceed a critical value (∼π⁻¹). The threshold wavelength of transmissibility (to a double-spaced coarse domain) is 5.2Δx, and waves longer than 6Δx are fully transmissible. Therefore, if l_c is gradually increased from 2 to 4 in a narrow zone abutting the interface, all waves approaching the interface are transmitted to the coarse domain without reflection. Thus, the present method meets conditions 1 through 3 quite well; in fact, it does much better than fourth- or even sixth-order difference schemes theoretically could. Furthermore, since spline bases are local, only three nodal amplitudes are needed to exactly satisfy up to three boundary conditions, either homogeneous or inhomogeneous, so that condition 4 can be easily formulated. Thus, the continuity of spatial fields at the interface will be identical to that in the domain interior, assuring the transmission of waves in both ways.

c. Time integration

The time integration is made in discrete time steps, Δt, by the leapfrog method or, alternatively, by the second-order Adams–Bashforth method.² For computational stability, either method limits the Courant number approximately to

View Expanded

where c represents the maximum wave speed possible in the model physics. The atmosphere we consider is compressible and contains acoustic waves as a possible mode of solution, even though acoustic waves carry very little information of meteorological significance. Thus, if the equations in section 4 are to be explicitly integrated, the size of a stable Δt will be severely restricted by the speed of sound. A number of methods are employed in atmospheric models to abate this acoustic problem. For example, by modifying the physical equations, the anelastic approximation (Ogura and Phillips 1962) eliminates the possibility of acoustic waves, or the quasi-compressibility approximation (Droegemeier and Wilhelmson 1987) slows down their speed; and the split time-level method (Klemp and Wilhelmson 1987) gains efficiency by calculating only the acoustic waves by simpler equations in small time steps.

We have decided to adopt the semi-implicit method. It does not remove acoustic waves, but slows down phase speeds in such a way that they are no longer a factor in determining stability. The method was originally proposed by Robert (1969, 1979) and now widely used in hydrostatic global models (against gravity waves); the first application to a nonhydrostatic model was made by Tapp and White (1976). In adaptation to the nested spectral model, we follow Burridge's (1975) formulation in which the time integration in each domain is split into two steps. In the first step, all the prognostic variables are explicitly predicted by the leapfrog (or Adams–Bashforth) method for Δt that is small enough only for gravity waves and other meteorological modes of solution. The explicit results, then, are “implicitly” adjusted in the second step. The required adjustments are determined by solving a second-order elliptic equation with a forcing term that comprises the second-order time difference of predictions at the current (i.e., the explicit step) and two previous time levels. Since the goal is not to precisely calculate acoustic waves but only to stabilize them, the adjustments need to be made only among the velocity components and the dry-air density, leaving the moisture variables unaltered.

In actual calculations, the explicit prediction is made in the geometrical space; the result is transformed to spline amplitudes; and the adjustments are made exclusively in the spectral space of amplitudes. The adjustment equation is actually an algebraic equation that is derived by applying the variational principle to the corresponding equations in the geometrical space. Since no cross differentiation is made in the derivation, there is no need for extraneous boundary conditions on vorticity or divergence. Furthermore, the algebraic equation can be efficiently solved by iterating a one-dimensional solver in alternate directions; two to three iterations in each direction are found to be sufficient. As the result, we have been able to increase Δt tenfold in exchange for a 10% increase in computation time per time step.

Due to the need for an extensive introduction of spectral notation, the mathematical detail of the algorithms is deferred to another paper which will be devoted to our numerical method.

a. The model setup

The simulated growth of a cloud cell in the two-dimensional vertical plane is discussed in this section. The ground is flat and no rotation (f = 0) is assumed. The cloud begins as a warm bubble near the ground and rises by its buoyancy through a horizontally stratified atmosphere, which is initially at rest. The background profile of the specific entropy, s, is shown in Fig. 1. The solid curve is the assumed background, which is a slightly humidified version of Jordan's (1958) mean tropical atmosphere for the hurricane season (the dotted line). The entropy of saturated air at the same temperature is shown by the dashed line, indicating the air below 2 km as potentially unstable if lifted. For the convenience of traditional interpretation, the diagram is also ruled by equivalent potential temperature, θ_e, which is calculated by a simplified definition,

θ_esc_Pa(6.1)

The initial disturbance is defined, in terms of temperature deviations, by

TT_maxπr(6.2)

where, for |x| ≤ a and 0 ≤ z ≤ (b + c),

View Expanded

and a = 16, b = 3, c = 0.5, all in km, and ΔT_max = 3 K. The relative humidity is not changed in the initial disturbance.

The domain configuration of the model is very flexible. To maintain some order in the presentation, the results shown in this section use the same, concentric nesting to five levels. The innermost domain is in |x| ≤ 24 km, and the size doubles at the next level of nesting, as is shown in Figs. 1 and 2. The outermost domain extends to x = ±384 km where periodicity closes the domain. While the domain sizes are fixed, the horizontal resolution varies with each experiment. Thus, Δx of the innermost domain, ranging from 200 m to 4 km, may identify a particular run; and, in any run, Δx of the other domains are outwardly doubled.

The vertical extent of the modeled space is from 0 to 21 km. The vertical resolution is fixed at Δz = 500 m, except for the case of Δx = Δz = 200 m. A condition w = 0 is enforced at the top and bottom boundaries. For other prognostic variables, the second derivatives w.r.t. z are set to zero at the bottom, while both the first and second derivatives are zero at the top.

The time integration employs the leapfrog/semi-implicit combination; in the innermost domain, Δt is 1.25 and 2.5 s for Δx = 200 and 500 m, respectively, and 5 s for other Δx; and it may be outwardly doubled in the outer domains, although it should be kept under 40 s for which the vertical propagation of gravity waves becomes unstable when Δz = 500 m.

b. Propagating waves

Before focusing our attention on the main event near the origin, we briefly discuss propagating waves in outer domains.

While the background state is hydrostatically balanced, the initial bubble is not, and it explodes like a little bomb, creating acoustic waves. Most of them reverberate between the top and the bottom of the domain and lose identity as recognizable waves. Only one component, known as the Lamb wave, propagates horizontally for a long distance. Figure 2 depicts the wave 10 min after the initiation, located at 180 km from the origin. Thus, the wave has been propagating at a speed of 300 m s⁻¹. The thick ticks on the x axis mark the location of domain interfaces, and Δx for each domain is also noted. The wave has already crossed the interfaces at x = 24, 48, and 96 km, and is crossing another at 192 km. Although amplitudes are small, the u and p perturbations are vertically in phase, and w is very small; in an isothermal atmosphere w would exactly vanish.

The rising bubble at the origin also generates gravity waves. The leading wave of the first internal mode is depicted in Fig. 3, at t = 1 h and x = 85 km. The wave was not fully formed until the bubble rose to the top in about 30 min, so that the propagation speed of the wave is about 50 m s⁻¹. Both the u and p perturbations have opposite signs between the upper and lower levels of the troposphere, while w is maximum at mid levels and lags in the horizontal phase. Thus, the wind field has the appearance of clockwise rotation as it propagates to the right. The wave has already crossed two interfaces, will cross another at 96 km, and is followed by slower gravity waves of higher modes.

It may be noted that our interface condition for nesting is not selectively adjusted to a certain mode or speed of waves, but simply asserts that the order of spatial continuity of fields at an interface is identical to that in the interior of any domain. Thus, it works equally well with acoustic and gravity waves, shown above and with advection, shown later.

c. The growth of a single cloud

The growth of a cloud in the 500-m resolution run is shown in Fig. 4, with 12 panels at 5-min intervals for 1 h. Since the growth is symmetric in x, only the right side is depicted. In each panel, the specific entropy is shown by solid contour lines, the cloudy areas (ρ_c ≥ 0.02 g m⁻³) are outlined by a dotted line, and the precipitation water content is in gray shades (7 levels for ρ_r ≥ 0.1, 0.5, 1, 2, 3, 4, and 6 g m⁻³). This graphical convention for ρ_c and ρ_r is used in all of the subsequent figures. Wind fields are shown later.

Although the initial perturbation (6.2) is rather broadly defined in x, the rising bubble, as seen in Figs. 4d,e, is very compact, with a sliver of warm³ air under the cap, and easily penetrates through the cold middle layer. In Fig. 4f the top has reached the tropopause, and a clockwise rotation becomes apparent at the edge of the developing canopy, which spreads farther out and becomes turbulent with time. After Fig. 4h, the thick canopy of “snow” gradually descends; and in Figs. 4k,l, a secondary cell is found near the bottom at x = 4 km, forced by a small descending tongue of cold air.

A similar sequence for the 2-km resolution run is shown in Fig. 5. The initial growth is similar but slower; the top rises to the tropopause (in g) 5 min later than in Fig. 4f. By comparison, however, most conspicuous is the absence of a turbulent appearance. The canopy is very thin, because a large portion of condensed water does not rise above 5 or 6 km but falls down early. These differences are analyzed below.

d. Precipitation at the ground

As is known from the linear theory of conditional instability, the vertical acceleration of an unstable parcel or plume depends on its horizontal scale. Such a scale in nonlinear simulations is dynamically determined and affected by the choice of spatial resolution. Figure 6 illustrates such effects by three pairs of diagrams, (a_1,2), (b_1,2), and (c_1,2), for Δx = 1, 2, and 4 km, respectively. In the first diagram of each pair, the vertical velocity, w (contours), and the precipitation water content, ρ_r (gray shades), both at the center of symmetry, x = 0, are plotted against time. The second diagram is the rate of precipitation at the ground, ρ_rW, also plotted against time.

The last pair (c_1,2) is the easiest to interpret. The initial bubble rises slowly; although w eventually reaches its maximum value of 21 m s⁻¹ at z = 10 km, it remains less than 10 m s⁻¹ below 5-km level, and much less at earlier times. The freezing level is about 5 km, and the terminal speed, W, of raindrops is typically 6 to 8 m s⁻¹, so that the precipitation actually falls downward almost as soon as it is formed. Thus, the ground receives a gentle rain continuously after 20 min. (Note that this is more than “drizzle,” since a rate of 50 g m⁻² s⁻¹ is 18 cm of water per hour). Only a small portion of condensed water is lofted above 6 km, due to the increasing w and the decreased W of “ice-phase” precipitation.

Similar interplays of these factors are involved in the other pairs of Fig. 6, but the greater intensity of w produces different precipitation patterns. In (b_1,2), the ground precipitation does not begin until a lull in w occurs at lower levels around 27 min. In (a_1,2), it is delayed until t = 36 min after the second pulse of updraft; an observer on the ground would see an ominous dark cloud gathering overhead for half an hour and, then, suddenly get doused in a torrential rain. In the runs with Δx = 200 and 500 m, the onset of ground precipitation is farther delayed and occurs more suddenly than those shown in Fig. 6, although diagrams for these runs are not shown because the rapidity of the w pulsation makes contours too closely packed. The maximum w is 31 and 45 m s⁻¹ in (b₁) and (a₁), respectively, and it reaches 60 m s⁻¹ in Δx = 200 m.

e. Wind fields

The wind fields associated with the rising bubble are shown in Fig. 7. Each of the first three rows (Figs. 7a–c) for Δx = 0.5, 1, and 2 km, contains three panels at t = 30, 35, and 40 min, while the panels in the last row (Fig. 7d), for Δx = 4 km, are at t = 45, 50, and 55 min, due to the slow rate of growth. In each panel, the precipitation water content is shown in shaded gray (as in previous figures), the cloud areas are outlined by a solid curve, and the wind vectors in the vertical plane are shown by arrows. The direction of the arrows is correctly adjusted for the scales on the x and z axes, while their length is not linearly but only monotonically related to the speed in order to accommodate large variations in a tight space.

In all the cases, a rotary circulation develops at the edge of a mushroom-shaped canopy of cloud that is associated with the formation of the first gravity wave shown in Fig. 3. In the lower resolutions, Figs. 7c,d, the flow is largely laminar; and the main circulation splits into two parts, the rising upper part and the rain-loaded lower part. In the higher resolutions, Figs. 7a,b, the circulation is far more complex, reflecting interactions between the initial buoyancy, gravity waves, and precipitation; the vertical motion even at the center is not simply up or down, and the flow breaks up into smaller eddies especially in the spreading canopy.

This trend toward complexity is depicted in Fig. 8 for Δx = Δz = 200 m in three panels, (a₁) to (a₃), at t = 31, 33, and 35 min. The wind vectors are plotted only on the right side of each panel to afford an unobstructed view of condensed water on the left. Due to reduced fall speeds, the ice-phase condensate is a good tracer of delicate flows in the canopy.

In (a₁), a pair of mushroom-shaped protrusions has developed on the top of the main canopy; and the rotation at the outer edge of the canopy is concentrated in a mamma-like hanging protrusion. In (a₂), the rotary circulations associated with the little mushrooms grow stronger, intensifying the narrow downdraft between them (at x = 0), while the large canopy almost splits into two parts on each side with two hanging protrusions that rotate in the same direction. In (a₃), only 4 min after (a₁), the entire canopy is a mass of small eddies. The behavior of the second pulse of the updraft at the center of symmetry is also interesting. In (a₁), it is stopped at z = 7 km, meeting a downdraft from above, and splits sideways in (a₂) and (a₃), generating its own eddies.

Grabowski and Clark (1991) have discussed the cloud-top instability that generates eddies resembling those in Fig. 8, although only one mushroom grew at the center. In their model, the anelastic approximation was used, the cloud did not precipitate, and the spatial resolution was typically 5 m. We are quite tempted but have not yet run our model at such a high resolution.

f. Asymmetry, natural and artificial

The narrow central downdraft at the top and the collision of the up- and downdrafts in the middle, both in Fig. 8, are singular events due to the assumed strict symmetry. In contrast, the results of two asymmetric simulations are shown in Fig. 9: in Fig. 9a, a weak vertical shear, 5 m s⁻¹ per 10 km of height, is assumed in the initial state; in Fig. 9b, there is no shear but the initial bubble is placed exactly at the interface of two domains in such a way that Δx = 200 m on the positive side of x and 400 m on the negative side. Otherwise, both should be compared with Fig. 8 (a₂).

In Fig. 9a with shear, the two little mushrooms are slanted and no sharp downdraft is found between them; and the collision of the up- and downdrafts at 7 km is diffused. In Fig. 9b the case of interface straddling, only one of the little mushrooms grows at the top, and the opposing drafts in the middle slide past each other. Apart from these details, the general appearance of the two results, including the degree of asymmetry, is quite similar.

As a matter of fact, the similarity is fortuitous, since the asymmetry in Fig. 9a is due to a natural cause while that in Fig. 9b is an artifact of modeling. However, the latter is not a simple numerical problem such as computational dispersion, which is practically absent in the present model. It is rather a fundamental problem of the finite resolution that places a limit on the realizable extent of dynamics in a numerical model. Earlier in this section, the dynamic effects of resolution were examined in separate runs. Fig. 9b demonstrates the degree of inevitable distortion that occurs when the resolution jumps in the midst of a severe convective event.

a. Anatomy of a simulated squall line

The configuration of nested domains is also similar to the earlier one, but each nested domain is twice as wide, slightly off-centered, and on a relative x coordinate translating at the constant speed U₀. Specifically, the innermost domain lies within (−24, 72) km at t = 0, will be within (12, 108) km after 1 h, and so on. All the outer domains translate at the same speed. The fifth and periodic outermost domain is 1536 km wide, so that even the fastest gravity wave will take 8.5 h to come around to affect the central area of interest.

A snapshot of the well-organized squall line at 5 h, 50 m in the 500-m resolution run is shown in Fig. 10a. The cloud areas (thin solid lines), precipitation (gray shades) and specific entropy (medium and heavy solid lines) are shown in the same manner as in earlier figures, but the plotted winds (arrows) are relative to the moving coordinate (by subtracting U₀). The x axis is labeled in the true distance from the point of the initial disturbance, and the interface originally at x = 72 km is presently at 282 km, where the resolution changes from 500 m on the left to 1 km on the right. The squall system has been running mostly within the moving, innermost domain but is now overtaking the interface. The vertical profiles of u (velocity) and s (entropy) are shown in Figs. 10b, c, respectively, at x = 260, 290, 320, and 330 km, each marked by a triangle in Fig. 10a. For reference, the initial profile of either u or s is also shown by a dash–dot line. The propagation speed of the system at this time is 19 m s⁻¹.

From Fig. 10a and (c_1–3), it is evident in the precipitating area that a shallow wedge of cold air (cooled by up to 7 K in temperature) has accumulated above the ground. A jet of the cold air with a peak at z = 2 km is found in (b₁); it descends to the ground (b₂) and accelerates to a speed greater than 30 m s⁻¹ in (b₃) as its depth decreases. Beyond the leading edge of this gust front at x = 322 km, no sign of the gust is found in (b₄), and the entropy profile in (c₄) is hardly modified from the initial state. The advancing cold wedge lifts the hitherto undisturbed air and initiates incipient cloud cells, of which some may fizzle but many grow into mature cells and join the main body of the convective system. The entropy maxima in (c_1–3) are the signature of those rising cells that have been left behind the advancing cold air.

At the top of the system, z ∼ 11 to 14 km, the downshear outflow of 20 m s⁻¹ carries the precipitating cirrus canopy forward, while the weaker upshear outflow leaves a long trailing cirrus canopy behind the system. The flow at midlevels is varied; it enters the system from the front, is partially caught in convective updrafts and downdrafts, and exits through to the rear.

The pressure field (not shown) contains high-frequency small-scale fluctuations, reflecting the continual but intermittent activities of numerous cells. Apart from these fluctuations, the surface pressure under the system shows a conspicuous high plateau of p′ ∼ 2.5 hPa between x = 250 and 320 km with a slight average gradient which is consistent with the acceleration of the surface wind.

b. Generation of squall lines

Once a simulated disturbance gets its various parts organized in a certain configuration, such as shown in Fig. 10, it becomes a self-perpetuating system as a squall line. On the other hand, the way such organization develops from the assumed initial bubble is a process that requires a favorable combination of many factors. By setting the evaporation rate of precipitation to zero, we can confirm the essential role of evaporative cooling in the generation of a squall line. However, if the evaporation rate is increased without adjusting the fall speed of precipitation, the cooling takes place at wrong places for the creation of cold air above the ground. The preexistence of a low-level wind shear favors a well organized squall line, especially its maintenance after formation. However, the role of shear in the formation is rather ambivalent; a propagating line of convective clouds can form without preexistent shear; and a strong shear may delay, or even prohibit, the organizing process.

To examine this question about the role of shear, a series of experiments has been conducted in which U₀ of (7.1) is varied from 0 to 20 m s⁻¹ while z_sc is fixed at 1 km (except for one case mentioned later). This choice of U₀ (the wind aloft) as the primary designator of the boundary shear, rather than the vertical gradient of U(z), follows the convention adopted by Fovell and Ogura (1989, hereafter FO89), although our U₀ does not numerically correspond to their Δu due to the use of different profile functions. The configuration of nested domains is similar to the case discussed in the previous subsection, but the translation speed of domain interfaces has been adjusted for each case in order to keep, as much as possible, the propagating activities within the innermost domain of 192-km width.

The results are summarized in Fig. 11, in which the precipitation water content ρ_r at z = 0 is plotted in gray shades as a function of x and t for five cases of U₀ = 0, 5, 10, 15, and 20 m s⁻¹. The time origin for each case is offset to avoid overlap. The horizontal wind speed u at z = 0, also a function of x and t, is contoured in Fig. 12 for the same cases. The contour levels are chosen for a clear depiction of the gust front. The trace of a point moving at the speed of U₀, starting at the origin of each case, is shown by a dashed line.

In the case of U₀ = 0, the cold pool of air accumulates under the initial cell for the first hour and then starts spreading out at a speed of about 10 m s⁻¹, initiating weak convective cells on its path in a quasi-periodic fashion. The wind u locally exceeds 20 m s⁻¹, but such episodes are intermittent and do not attain the structure of a gust front. Nevertheless, the cloud system represents a clearly defined line of passing showers.

The tendency to form a lasting gust front is enhanced by an increased U₀, but only to a certain limit. In the case of U₀ = 5 in Fig. 12, a continuing zone of u > 20 m s⁻¹ is formed behind the rapidly advancing front after t = 3 h. In the case of U₀ = 10, a zone of u > 30 m s⁻¹ develops behind the front, and the peak values exceed 40 m s⁻¹ just at the front. The precipitation patterns in Fig. 11 indicate regular regeneration of intense new cloud cells in these cases. The trend appears to be reversed at U₀ = 15; although new cells are haphazardly initiated, they simply drift with the wind aloft for the first 5 h, leaving a passive wake of cold air behind them. A weak gust front is eventually formed after 6 h, however. In the case of U₀ = 20, there is no such possibility since convective activities cease after 6 h.

The propagation speeds of simulated squall lines for various U₀ are depicted in Fig. 13. A triangular symbol indicates the speed that is measured by the slope of a gust front in (x, t) plot after it is well established. The case of no shear is included in this category since it appears to be a smoothly connected limit of sheared cases. Two symbols plotted for the same value of U₀ indicate a degree of uncertainty either in the speed itself or in the way of measurement. In spite of this uncertainty, however, the propagation speed seems to reach a maximum at approximately 21 m s⁻¹ between U₀ = 10 and 12.

At U₀ = 17 and 20, a “+” symbol indicates the speed of drifting clouds, since no gust front or squall line has formed. It was reported by FO89 that “storms” (squall lines) also failed to form for Δu > 22.5 when the model simulation was started from an initial bubble, but that the storm continued to propagate, if Δu was gradually increased from 22.5 to higher values after the storm was established at the low value. Thus, their 22.5 appears to correspond to our threshold at U₀ = 15. If we had employed a similar maneuver to increase U₀ during a run, we might have found squall lines propagating at higher speeds than 21 m s⁻¹. Then, the present maximum in Fig. 13 and a slight irregularity found in a similar diagram (Fig. 2) of FO89 may be related phenomena, although many differences in the experimental designs make such a comparison imprecise.

The depth of the sheared layer is one such difference; it is 2.5 km for a linear wind profile in FO89. In order to reduce the gap, we have conducted one experiment with z_sc = 2 km and U₀ = 10 m s⁻¹ for our hyperbolic-tangent profile. In this experiment, an initial gestation period lasts for 6 h with haphazard convective activities before a well-organized squall line emerges. When it is finally established, however, the propagation speed is found to be nearly identical to the speed in the case of z_sc = 1 km, as shown by an asterisk in Fig. 13, suggesting that the depth of the sheared layer is not a sensitive factor in determining the speed of a mature squall line.

The effects of spatial resolution on the simulated squall line with U₀ = 10 are demonstrated in Fig. 14, for Δx = 0.5, 1, 2, and 4 km. The contour lines of horizontal winds are now superposed on the precipitation in gray shades. As the resolution becomes coarser, the gust front takes more time to develop and attains lesser intensity, although the propagation speed remains in the range of 18 to 20 m s⁻¹. The precipitation patterns are more sensitive to the resolution. Especially in the case of Δx = 4 km, the formation of new cells is infrequent and each cell is extremely broad.

Weisman et al. (1997) presented similarly broad structures of squall lines, simulated in various resolutions from 1 to 12 km. In their experiments, an initially assumed cold pool of air initiates convection, while, in the present experiments, the cold pool has to be created by convection. We do not know at this time whether or not a squall line can bootstrap itself with 12-km resolution.