a. Governing equations

The material time derivative expressed in the generalized vertical coordinate η is written as

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where the partial time derivative and the horizontal-gradient operator ∇ are evaluated on constant-η surfaces, η̇ ≡ Dη/Dt is the generalized vertical velocity, and v is the horizontal velocity.

The mass continuity equation is expressed in terms of the pseudodensity m as

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where

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Here, ρ is density and z is geopotential height. The upper and lower boundaries are assumed to be material, constant-η surfaces, so the boundary conditions are

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where the subscripts T and S refer to the upper and lower boundaries, respectively. The horizontal momentum equation is given by

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where f is the Coriolis parameter, k is the vertical unit vector, HPGF is the horizontal pressure-gradient force vector, and F is the horizontal friction force. The horizontal pressure-gradient force may be written in terms of pressure (p) as

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or in terms of the Exner function (Π) as

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where g is gravity, ϕ = gz is the geopotential, the Exner function is defined by

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and θ is the potential temperature given by

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where T is temperature. The vertical momentum equation is

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where w is the vertical velocity, VPGF is the vertical pressure-gradient force, and F_z is the vertical component of the friction force. The vertical pressure-gradient force may be written in terms of pressure as

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or in terms of the Exner function as

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The lower boundary condition on w is given by

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The thermodynamic energy equation can be written in terms of the potential temperature as

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where Q is the diabatic heating. Within the generalized vertical coordinate framework, geopotential height is a dependent variable. Using the definition w ≡ Dz/Dt, we can write the tendency equation for z as

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The atmosphere is assumed to be an ideal gas for which the equation of state is given by

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where R is the gas constant for dry air.

We define a terrain-following, height-based vertical coordinate σ as

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which is a variant of the coordinate introduced by Gal-Chen and Somerville (1975).

Neglecting diabatic heating and frictional forces, the set of equations given by (2.2)–(2.15) is one equation short of being a closed system. The remaining relation to be specified is that for the dependent variable η̇, the generalized vertical velocity, which is discussed in the following subsection.

b. Specification of the vertical coordinate and diagnosis of the generalized vertical velocity

1) The vertical coordinate

The above system of governing equations can be closed by specifying the vertical coordinate as a function of any combination of the dependent variables, as long as that function is monotonic in height. This leads to the determination of the generalized vertical velocity η̇. For example, choosing η = z leads to η̇ = ż = w, while choosing η = θ leads to η̇ = θ̇ = Q/Π. We employ an alternative approach, which is to reverse this process by specifying the generalized vertical velocity, and apply it in the vertical advection terms of the prognostic equations. As a result, η is not necessarily defined in terms of the dependent variables, yet it still serves as the vertical coordinate as long as it remains monotonic in height. This condition is met as long as the height field, determined by (2.13), remains a monotonic function of η. If this condition is violated, then the generalized vertical velocity may be respecified to maintain monotonicity. This approach is reminiscent of the ALE method of Hirt et al. (1974). Within the ALE framework, the generalized vertical velocity would be specified as zero until the height field becomes nonmonotonic, at which point an adjustment would be made. Lin (2004) employs a similar method.

As a further example of the approach described above, if η were initially specified as the quasi-Lagrangian isentropic coordinate, then choosing η̇ = Q/Π would automatically satisfy ∂θ/∂t = 0 through (2.12), which maintains θ = η. Similarly, if η were initially specified to be the geopotential height, then choosing η̇ = w would satisfy ∂z/∂t = 0 through (2.13), maintaining z = η. In the first example, the vertical coordinate is the “target value” for the potential temperature, while in the latter, it is the target value for the height.

The vertical coordinate in our model is based on the hybrid σ–θ coordinate developed by KA97. It is the terrain-following σ coordinate near the surface and transitions to θ coordinates with height. The coordinate serves as a target field for a specified function F(θ, σ). Therefore, the generalized vertical velocity diagnosis primarily follows KA97, but is modified in order to maintain coordinate monotonicity as well as specific requirements on the spatial distribution, or “smoothness,” of coordinate isosurfaces, to be described below. The maintenance of monotonicity near the surface is provided by the σ contribution to the vertical coordinate, while in the free atmosphere, where the coordinate is primarily θ, monotonicity is maintained by an adaptive grid method (e.g., Skamarock 1998; He 2002; Zängl 2007). While this method is typically associated with the discrete grid framework, here we formulate it in the continuous system.

The target relationship for the variables θ and σ is

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where the functions f (σ) and g(σ) are chosen such that

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The form of g(σ) that we use in the model is

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where r is a constant greater than unity. This choice satisfies (2.17), and the thickness of the σ-like domain near the surface can be controlled by the value of r: the larger its value, the nearer the surface the coordinate becomes fully isentropic. In KA97, the requirement ∂F/∂η > 0 is satisfied by choosing f (σ) from

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where θ_min and (∂θ/∂σ)_min are suitably chosen parameters representing the lower bounds of the potential temperature and static stability, respectively. In the model, we specify (∂θ/∂σ)_min = 0. This means that the condition ∂F/∂η > 0 is not necessarily met for statically unstable profiles. However, unlike KA97, in which η ≡ F, we allow F to depart from its time-independent target value η in order to maintain the requirement ∂η/∂σ > 0, as shown below.

The time tendency of F is given by

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where τ_rel is a relaxation time constant, and η̇_S is the smoothing contribution to the generalized vertical velocity to be described below. The first term on the rhs of (2.20) is a Newtonian relaxation term that relaxes F toward its target value η. The second term is an advective adjustment to F required to satisfy two smoothness criteria for coordinate isosurfaces, as described below. Note that in KA97, for which η = F by definition, the rhs of (2.20) is identically zero.

We prescribe two smoothness criteria for the spatial arrangement of coordinate isosurfaces: a horizontal criterion to maintain

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and a vertical criterion to maintain

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where the ()_max values in the above equations are specified maximum limits. The horizontal criterion (2.21) is designed to limit the existence of sharp horizontal gradients and their associated truncation errors in the discrete model. The vertical criterion (2.22) eliminates the possibility of η from becoming nonmonotonic with height, and in the discrete model, it prevents the relative difference in the thickness of adjacent layers from becoming too large. It basically serves to keep the distribution of layer thicknesses in a model column evenly distributed. This can be seen by considering a representative continuous relationship between z and η, as shown in Fig. 1. Three points are shown along the curve that represent discrete model positions. The relative difference in the thickness of adjacent layers is expressed by the nondimensional parameter

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where the δ² operator refers to the difference operator δ recursively applied twice. Applying a Taylor series expansion to (2.23), we obtain

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where the subscript “2” denotes continuous derivatives at the discrete point “2”, (δη)_A ≡ η₂ − η₁, and (δη)_B ≡ η₃ − η₂. For (δη)_A = (δη)_B = (δη), and truncating the Taylor series, we have

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which is the basis of the vertical smoothness parameter in (2.22).

In short, the formulation of the vertical coordinate involves two groups of “tunable” coefficients: 1) the KA97 parameters r, θ_min, and (∂θ/∂σ)_min; and 2) the smoothness parameters associated with the adaptive grid method. Optimal values for these mainly depend on the spatial scales of the atmospheric phenomena under investigation (which drive the choice of the smoothness parameters), the height above the surface above which the coordinate becomes isentropic, the time scale to return to the target value of F(θ, σ), and the expected lower bound for the potential temperature in the domain. So far we have tested the model in limited-area case studies, and it provides good results for a wide range of parameter values. However, when the model is extended to the global domain, the optimal values will vary in space and time. It may then be desirable to add the flexibility of having the parameters vary in order to optimally represent the various atmospheric regimes encountered globally and seasonally.

2) Diagnosis of the generalized vertical velocity

Equation (2.20) serves as the starting point for the diagnosis of the generalized vertical velocity. Note that when the rhs of (2.20) is zero, the vertical velocity has the same form as in KA97. For convenience, we express the generalized vertical velocity as the sum of two contributions; that is,

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where η̇_T is the “target seeking” contribution that maintains F(θ, σ) at (or relaxes it toward) its target value η and η̇_S is the smoothing contribution, introduced in (2.20), which provides the advective adjustment needed to maintain the spatial smoothness of the coordinate isosurfaces.

(i) “Target seeking” contribution to the generalized vertical velocity

Considering η̇_S equal to zero in (2.20), and applying the chain rule of differentiation to the lhs of (2.20), we can write

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Combining (2.12), (2.13), (2.15), and (2.27), and solving for the generalized vertical velocity, we obtain

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where

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which is the column height, and we used

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For F = η, (2.28) expresses the generalized vertical velocity of KA97. When F ≠ η, (2.28) has a singularity for ∂F/∂η = 0, which occurs in neutrally stable environments and F ≈ θ. Therefore, we must modify the generalized vertical velocity to avoid this singularity. A straightforward method is to “freeze” the coordinate isolines in space, such that ∂z/∂t = 0, as ∂F/∂η approaches zero, as well as for ∂F/∂η < 0. In other words, the coordinate becomes a stationary, Eulerian coordinate in regions of negative static stability. From (2.13), we then write

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For ∂F/∂η ≥ β, we use (2.28), where β is a parameter with a value between 0 and 1. For 0 < ∂F/∂η < β, we use a linear combination of (2.31) and (2.28) evaluated with ∂F/∂η = β; that is,

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The result is

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(ii) Smoothing contribution to the generalized vertical velocity

The generalized vertical velocity contribution required to advectively adjust the geopotential height field, per the horizontal and vertical smoothness criteria described above, is calculated from

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where the two terms in brackets are the height tendencies due to horizontal and vertical smoothing.

The horizontal smoothing tendency is quantified in the form of a “del-4” diffusion equation given by

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where κ_horiz is a constant diffusion coefficient. The smoothing tendency is nonzero only when |∇⁴z| > (∇⁴z)_max. The vertical smoothing tendency is similarly expressed using second-order diffusion. The vertical smoothing tendency is given by

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where κ_vert is a constant diffusion coefficient. Vertical diffusion only acts when the absolute value of the ratio of the second and first derivatives of z with respect to η exceeds the specified limit.

c. Vertical flux of horizontal momentum in a generalized vertical coordinate

In this section we derive a time tendency equation for the zonally averaged zonal velocity for the purpose of diagnosing the vertical momentum flux in the model. This is done to test the model’s accuracy and to contrast the form of the fluxes within the σ- and hybrid-vertical coordinate frameworks. In the derivation, we determine an expression for the Eliassen–Palm (EP) flux in a generalized vertical coordinate. The derivation follows Andrews (1983), who derived the EP flux in isentropic coordinates for quasi-static flow.

Combining (2.1), (2.3), (2.5), and (2.6a), the zonal momentum equation can be written

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where u and υ are the zonal and meridional velocity components, respectively, and F_x is the zonal component of the friction force. Combining (2.37) with the continuity equation (2.2), the zonal momentum equation in flux form can be written as

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The zonal average of a given property a may be defined as

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where L is the length of the domain in the zonal direction. Applying (2.39) to (2.38), we obtain

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where we used

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In a similar manner, the zonally averaged continuity Eq. (2.2) is

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Now divide the fluid properties into mean and perturbation components so that

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where the prime notation represents perturbations from the mean. Under Reynolds averaging, we have

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and

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Combining (2.40) and (2.42), and applying (2.45), we can write the zonally averaged zonal momentum tendency equation as

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Following Andrews (1983), the mass-weighted “residual” mean velocities are defined as

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and

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Applying (2.47) and (2.48) in (2.46), we obtain

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where F^(η) ≡ [0, F_y^(η), F_η^(η)] is the EP flux vector in generalized vertical coordinates, which has the meridional and vertical components

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and

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respectively. Equation (2.49) shows that the EP flux is nondivergent for steady-state, uniform, frictionless flow.

The vertical component of the EP flux given by (2.51) is the vertical flux of the horizontal momentum. In z coordinates, the first term on the rhs is zero, which leaves the eddy flux term − as the means of vertical momentum transport. In θ coordinates, for adiabatic conditions, η̇′ = θ̇′ = 0, which means the vertical momentum transport occurs through the first term on the rhs of (2.51), that is, the pressure form drag on coordinate (material) surfaces. This is consistent with the fact that the zonal average of the “mountain torque” term in [(A.1), see the appendix], which is the pressure form drag on the lower (material) boundary, represents the momentum flux across the earth’s surface. Klemp and Lilly (1978) also noted the equivalence of the pressure drag and momentum flux in isentropic coordinates.

a. Vertical grid

The vertical staggering of the prognostic variables of the model, shown in Fig. 2, is based on the Charney–Phillips (CP) grid (Charney and Phillips 1953) developed for hydrostatic p-based coordinate models. This grid is also the basis of the generalized vertical coordinate model of KA97. With the CP grid, the potential temperature is staggered with respect to the horizontal velocity. Another commonly used staggering is the Lorenz (L) grid (Lorenz 1960) in which the potential temperature is carried at the same levels as the horizontal velocity. The advantages of the CP grid over the L grid in quasi-static modeling have been analyzed in various papers (e.g., Arakawa and Moorthi 1988; Arakawa and Konor 1996). Thuburn and Woollings (2005), Toy and Randall (2007), and Toy (2008) have pointed out similar advantages to the CP grid in nonhydrostatic z-based coordinate models. These include the avoidance of vertical computational modes as well as improved accuracy in the representation of vertical wave propagation.

The grid indexing used in the vertical discretization is shown in Fig. 2. There are K layers in the model numbered from bottom to top. Layer centers are numbered by whole integers, k, and layer edges by half-integers.

b. Continuity equation

The discretized form of the continuity Eq. (2.2) is given by

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where the difference operator δ is defined as (δ a)_k ≡ a_k+1/2 − a_k−1/2. Since the top and bottom model boundaries are material as well as coordinate surfaces, the boundary conditions are

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Note that the model conserves mass due to the flux form of the continuity Eq. (3.1).

c. Pressure-gradient forces

From (2.10b), the discrete VPGF is given by

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and

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where the layer-center geopotential is interpolated as

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From (2.6b), the discrete horizontal pressure-gradient force (HPGF) is given by

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where

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and

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In the appendix, we show that the averaging used in (3.5) and (3.9) is necessary to satisfy an integral constraint on the total energy conservation.

The upper boundary is assumed to be a constant-height surface; therefore, w_K+1/2 = 0. The vertical velocity at the lower boundary is diagnosed from

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which comes from the boundary condition (2.11). Using (3.10), the horizontal and vertical tendencies are related by

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From (2.9), we express the tendency of w_1/2 as

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As mentioned above, w_1/2 is a diagnosed quantity; therefore, (3.12) is not used as a prognostic equation in the model. Instead, it is used to diagnose the surface pressure. Equations (3.4), (3.7), (3.11), and (3.12) can be solved for the unknown (VPGF)_1/2 (see Toy 2008). The surface pressure is then diagnosed from (3.4) by way of the surface Exner function.

d. Thermodynamic energy equation

The vertically discrete prognostic equation for θ is based on (2.12), except with the vertical advection term expressed as (mΠ)⁻¹[∂(mη̇Πθ)/∂η − θ ∂(mη̇Π)/∂η]. This term is written in centered form in the discrete equation, which is given by

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and

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where

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and

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Note that this form of the thermodynamic equation is similar to that of KA97. It is dictated by the CP-grid staggering and total energy conservation considerations discussed in the appendix.

e. Geopotential tendency equation

The tendency equation for the geopotential is written as

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where the layer-edge horizontal mass fluxes are given by (A.12) in the appendix.

f. Diagnostic relations

The density at layer centers is diagnosed from the pseudodensity and geopotential using the discrete form of (2.3) given by

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The diagnostic equation for temperature at layer centers is obtained from (2.8) and is based on the potential temperature interpolated to layer centers. It is written as

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The ideal gas law, which relates layer-center state variables, is written as

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and the Exner function is given by

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Given the density from (3.19), the temperature, the pressure, and the Exner function are solved simultaneously from (3.20)–(3.22).

g. Diagnosis of the generalized vertical velocity

The generalized vertical velocity is diagnosed through a multistep process. The first step is to calculate the target-seeking contribution η̇_T from the discrete forms of (2.28), (2.31), and (2.33), given by

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The next step is to calculate the smoothing contribution η̇_S which is based on the parameters (∇⁴z)_k+1/2 and (δ²z/δz)_k+1/2, given by (2.23). The discrete form of (2.34) is used, which is

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where, from (2.35) and (2.36), we use

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and

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Note that the smoothing contribution is generally zero except where it is necessary to advectively adjust the position of coordinate surfaces to maintain the smoothness criteria.

The final steps are to predict provisional values of the potential temperature and geopotential height based on the sum of the vertical velocity contributions given by (3.23) and (3.24). Following this, the function F (θ, σ) is calculated using these provisional values, and it is compared to the target F field for the current time step. A final vertical advective adjustment is then calculated to bring F to its target through an iterative procedure similar to that used by KA97. We refer to the generalized vertical velocity contribution from this adjustment procedure as η̇_ADJ. It is generally a small, residual contribution that results from the vertical discretization and reduces to zero for infinite resolution.

The provisional values for θ and z are calculated from

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and

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where the superscript asterisks (*) refer to the provisional values, n refers to the current time step, and Δt is the time-step length. Euler forward time stepping is shown for simplicity. From these values, we calculate

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where

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The target value for F at time step n + 1 is calculated from the prognostic equation

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which is based on (2.20) with η̇_S set to zero. Equation (3.31) is designed to relax toward the vertical coordinate value η_k+1/2. The difference between the provisional and target values of F is then defined as

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The purpose of (η̇_ADJ)_k+1/2 is to bring (Δη)_k+1/2 to zero, which, at each iteration, is solved from

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The result is used to update θ* and σ* from

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and

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Equations (3.29) and (3.32)–(3.35) are iterated until Δη becomes sufficiently small. The final value of (η̇_ADJ)_k+1/2 is the cumulative value calculated in each iteration. The resulting values of θ* and σ* are the final n + 1 values.

Finally, the generalized vertical velocity is given by

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Note that in the continuous limit, η̇_ADJbecomes zero, and (3.36) reduces to (2.26).

a. Small-amplitude mountain waves in an isothermal atmosphere

A simple test case is presented to demonstrate the ability of the model to represent small-amplitude gravity waves. Smith (1979) provides a review of mountain-wave theory that includes linear analytical solutions with which the model results can be compared. In the experiment, the flow is initially isothermal with a temperature of T = 287K and is purely horizontal with zonal wind speed u = 20 m s⁻¹. (Overbars represent the basic state.) The buoyancy frequency, given by , a constant for isothermal atmospheres, is 0.0183 s⁻¹. Following Queney (1948), the mountain is an isolated barrier whose profile is prescribed as a witch of Agnesi curve given by

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where z_S(x) is the surface height, h is the mountain height, and a is the half-width. In this experiment, h = 10 m and a = 2 km. A measure of the linearity of the wave solution is the inverse Froude number based on the mountain height, which is Nh/u. Linear wave theory applies when the mountain is small, that is, when Nh/u ≪ 1. In this experiment, Nh/u = 0.00915 ≪ 1, so we can compare the model results to the analytic linear solution. The inverse Froude number based on the mountain half-width is given by Na/u. For broad mountains, in which Na/u ≫ 1, the flow is approximately hydrostatic. With narrow mountains, the hydrostatic approximation breaks down. In our case, Na/u = 1.83 ∼ 1, so the flow is nonhydrostatic. Figures 3a and 4a show the steady-state analytic linear solutions to the perturbation zonal and vertical velocity fields (u′ and w′), respectively. Note that phase lines tilt upwind, while wave packets have a downwind tilt characteristic of nonhydrostatic flow. The solution is based on the first 90 Fourier modes of the surface topography. The lateral boundaries are periodic with a domain length of L = 120 km, and radiative upper boundary conditions are applied.

In the model simulations, the horizontal boundary conditions are periodic and the domain length is L = 120 km. The upper boundary is a rigid lid at 30 km. A Rayleigh damping layer based on Klemp and Lilly (1978) is used in the upper portion of the domain. The model is run with 120 levels and 600 horizontal grid points. For the hybrid-coordinate run, we use θ_min = 270 K, (∂θ/∂σ)_min = 0 K, and r = 64. As shown in Fig. 5, this provides a rapid transition with height from the terrain-following coordinate to the θ coordinate. At z ≈ 3 km and above, the coordinate is nearly isentropic. Figures 3b and 4b show numerical model results with the σ coordinate, and Figs. 3c and 4c show results with the hybrid coordinate. Both simulations agree well with the analytical solution. For both simulations, the perturbation velocity fields are shown at time t = 40a/u (1.11 h), at which time the models reached an approximate steady state.

Linear theory provides an analytical solution to the vertical flux of horizontal momentum. Following Eliassen and Palm (1961), the momentum flux is written as

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which, for steady-state conditions, is constant in height. From the model runs, we diagnose the nonlinear momentum flux given by (2.51). We define two components of the momentum flux: the “eddy flux” component given by

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and the “form drag” component given by

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The surface pressure drag M_SD is equal to the form drag component evaluated along the surface; that is,

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For steady-state flow, the following relation theoretically applies at all levels:

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Figure 6 shows the vertical profiles of the eddy and form-drag contributions to the momentum flux at time t = 40a/u. The sum of these, shown by the black curve, theoretically equals the surface drag. The theoretical result is plotted as the vertical red line for reference. In each coordinate system, the total momentum flux is nearly constant with height and is close to the theoretical surface-drag value, as well as the linear momentum flux given by (4.2). The sign of the total momentum flux is negative, which means that the surface exerts a drag on the atmosphere, as expected. Note the nonzero contribution of the form drag in σ coordinates (Fig. 6a), which is due to the sloping of the coordinate surfaces with respect to z. In the Eulerian system, the vertical variations of the eddy-flux and form-drag contributions cancel in such a way as to keep their sum approximately constant in height. With the hybrid coordinate, a simpler view of momentum transport is afforded, as the flux is due entirely to the form drag component above ∼3 km. This is because the coordinate is almost purely isentropic, so the vertical velocity is zero and, therefore, the eddy flux is zero as well.

b. The 11 January 1972 Boulder, Colorado, downslope windstorm

The downslope windstorm that occurred in Boulder, Colorado, on 11 January 1972 was an extensively observed meteorological event (Lilly and Zipser 1972). It provides an ideal test case for model simulations of mountain-wave amplification and breaking. In this experiment, we follow Doyle et al. (2000), which presents various two-dimensional (x–z) simulations of the windstorm. The topography is idealized by a witch of Agnesi curve with the height and half-width set at h = 2 km and a = 10 km, respectively. The free-slip condition is applied at the lower boundary. The horizontal domain is 220 km in extent, and we use periodic lateral boundary conditions. The horizontal grid spacing is Δx = 1 km. The model top is a rigid lid at z = 48 km. In the lower 35 km, we use even vertical grid spacing in z. Above this level, the grid is stretched by gradually increasing the layer thickness up to the model top.

The baseline vertical resolution is 125 levels in the lowest 25 km, giving an average vertical grid spacing of 200 m. A high-resolution σ-coordinate run was also performed using 500 levels in the lowest 25 km. Three simulations are presented: 1) the “Sigma125” run using σ coordinates and the baseline vertical resolution, 2) the “Sigma500” run using σ coordinates and the high vertical resolution, and 3) the “Hybrid125” run using hybrid coordinates at the baseline vertical resolution. The hybrid vertical coordinate parameters introduced in (2.16), (2.18), and (2.19) were specified as θ_min = 270 K, (∂θ/∂σ)_min = 0 K, and r = 16. As shown in Fig. 7, the hybrid coordinate is primarily isentropic above z = 10 km.

In this experiment, coordinate surface smoothing is applied in the hybrid vertical coordinate runs to allow for isentropic overturning. Referring to (3.25) and (3.26), the maximum smoothness thresholds, above which smoothing occurs, are specified as (∇⁴z)_max = 3.4 × 10⁻¹¹ m⁻³ and (δ²z/δz)_max = 0.4. The diffusion coefficients used for horizontal and vertical smoothing are κ_horiz = 3.125 × 10¹² m⁴ s⁻¹ and κ_vert = 1000 m s⁻¹, respectively. On two-gridpoint length scales, these coefficients correspond to e-folding times of O(10⁻² s); that is, the smoothing acts on a short time scale. (This is on the order of the model time steps, which, for now, is limited by vertically propagating acoustic waves and the use of an explicit time-differencing scheme: third-order Adams–Bashforth.) The relaxation time constant introduced in (2.20) is specified as τ_rel = 0.5 h, which is on the order of the time scale on which the wave breaking occurs. Finally, the parameter β used in (3.23) is set to 0.7.

We apply a subgrid-scale mixing parameterization to the three components of velocity as well as the potential temperature, following the scheme used in the University of Oklahoma’s Advanced Research Prediction System (ARPS; documentation available online at http://www.caps.ou.edu/ARPS/download/code/pub/ARPS.docs/ARPS4DOC.PDF/arpsch6.pdf). We use the modified Smagorinsky first-order closure scheme (Smagorinsky 1963), which includes Richardson number dependency.

The initial conditions, shown in Fig. 8, are uniform in the horizontal following Doyle et al. (2000). They are based on the upstream 1200 UTC 11 January 1972 Grand Junction, Colorado, sounding up to 25 km. At higher levels, a constant zonal wind of 7.5 m s⁻¹ is assumed and the temperature profile smoothly merges with that of the U.S. Standard Atmosphere, 1976. The reference surface pressure corresponding to z = 0 is 850 mb.

The potential temperature field at t = 70 min is shown in Fig. 9. The mountain wave has substantially developed throughout the troposphere and lower stratosphere with generally an upwind tilt to the phase lines. A hydraulic jump feature has developed in the lower troposphere, approximately 20 km downstream of the mountaintop with lee waves of horizontal wavelength ∼10 km appearing just downstream. There is also considerable wave development above the hydraulic jump at the base of the stratosphere, with lee waves appearing just downstream as well.

The three simulations shown in Fig. 9 agree well with each other. The most noticeable difference is that in the σ-coordinate runs, isentropes are already beginning to overturn at the 19-km level. This overturning is more pronounced in the Sigma500 run. In the Hybrid125 run, overturning has not occurred yet, which is likely due to the decreased resolution of the isentropic coordinate in the areas of low static stability. Coordinate surfaces closely follow the isentropes above ∼10 km.

A particular advantage of the isentropic coordinate is the enhancement of vertical resolution in regions of high static stability. As shown in Fig. 10, which shows the static stability field at t = 2 h, this advantage leads to more accurately resolved layers of stable air. This was also recently noted by Zängl (2007), who found improved representation of the tropopause in nonhydrostatic simulations with the isentropic coordinate. The static stability field resulting from the Hybrid125 simulation (Fig. 10c) closely resembles that of the high vertical resolution Sigma500 run (Fig. 10a) in terms of capturing layers of high static stability. In the Sigma125 run (Fig. 10b), these layers are much less pronounced. Use of the hybrid coordinate, therefore, provides an economic advantage over the σ coordinate by capturing these features with the same number of model layers. On the other hand, Fig. 10 shows that the hybrid coordinate underestimates the degree of static instability in regions of wave breaking. As mentioned before, this is likely due to the reduced vertical resolution in these regions.

At 3 h of simulation time, the wave-breaking activity is at a maximum, as is the surface-wind intensity on the leeward mountain slope, that is, the simulated downslope windstorm. The potential temperature and zonal wind fields at this time are shown in Figs. 11 and 12, respectively. There is general agreement among the three model configurations. These results also compare well with the model results presented in Doyle et al. (2000; see Figs. 3 and 4 of their paper).

The diagnosed surface form drag, given by D = −∮_xp_S(∂z_S/∂x)dx, is plotted as a time series in Fig. 13. The evolution of the drag force is related to the downward transfer of zonal momentum due to the amplification and breaking of the mountain wave (e.g., Peltier and Clark 1979; Durran and Klemp 1983). As shown in Fig. 13, the three simulations produced similar results during the first 1.5 simulated hours, which is the period of wave amplification that precedes wave breaking. After this time, differences develop among the runs, but they generally follow a similar evolution. For the hybrid-coordinate simulation, this indicates that the process of wave amplification and breaking is adequately represented in terms of the vertical transfer of zonal momentum. After the peak at 3 h, the drag dies out and even becomes negative for a brief period. This is due to the periodic lateral boundaries and the fact that the momentum field is not forced.

The most striking difference between the hybrid- and σ-coordinate runs is in the vertical advection of a passive tracer. Here, we see a distinct advantage with the isentropic coordinate. To isolate the effects of vertical advection as much as possible, the passive tracer is initialized along horizontal bands bounded by selected isentropes as shown in Fig. 14a. The tracer is assigned the arbitrary value of unity inside the bands and zero outside. This is also shown in a scatterplot of tracer concentration versus potential temperature for all model points, as shown in Fig. 14b. In the continuous system of equations for adiabatic processes, since θ is conserved, the correlation between θ and the passive tracer remains unchanged in time assuming no diffusion of either property. This means that the scatterplot of tracer concentration versus θ should remain unchanged in time.

Profiles of the tracer concentration after 70 min of simulation time are shown in Fig. 15. In contrast to the initial conditions shown in Fig. 14a, there are now additional values of tracer concentration besides 0 and 1, and some of the tracer has “leaked” outside of the original isentropic bounds indicated by the boldface black curves. This has occurred because of numerical dispersion associated with the vertical advection terms of the tracer tendency equation. The dispersion error is most evident where the coordinate is σ, that is, in Figs. 15a and 15b and the lowest band in the hybrid coordinate plot in Fig. 15c. At t = 70 min, wave overturning has not yet occurred and there has been minimal coordinate smoothing in the hybrid vertical coordinate run. Therefore, the vertical velocity in the θ-coordinate regions of the hybrid coordinate has been virtually zero up until this time. The effect of this can be seen in the upper three tracer bands in Fig. 15c as compared to those in Figs. 15a and 15b.

The characteristics mentioned above are more noticeable in Fig. 16. The difference between the top tracer band among the four simulations is the most striking. With the hybrid coordinate (Fig. 16c), the scatter points lie along the theoretical profile indicated by the dashed lines. In the 125-level σ-coordinate simulation (Fig. 16b), the profile of the upper band differs significantly from theoretical profile due to dispersion error and numerical diffusion. The 125-level hybrid-coordinate model also has an advantage over the high-resolution 500-level σ-coordinate simulation (Fig. 16a), in which some dispersion is evident at the discontinuities in the original profile.

Figure 17 shows scatterplots after three simulated hours. The hybrid run displays some dispersion error due to the vertical velocity induced by coordinate smoothing. Despite this, the hybrid run exhibits less error than the Sigma125 runs. The error is comparable to, if not better than, the high-resolution Sigma500 runs, but achieves this with fewer model levels.