Adaptation of θ-Based Dynamical Cores for Extension into the Thermosphere Using a Hybrid Virtual Potential Temperature

Stephen D. Eckermann; Cory A. Barton; James F. Kelly

doi:10.1175/MWR-D-22-0320.1

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Fig. 1.

Solid curves show fitted one-dimensional NAVGEM profiles of (a) R, (b) κ⁻¹, and (c) c_p for F_10.7 ranging from 50 to 290 sfu. Dotted curves in (c) show implicit c_p values in the standard HVPT dynamical core: c_p = (7/2)R.

View raw image

Fig. 2.

Schematic representation of limits of hybrid virtual temperature $T_{υ_{h}}$ : a moist well-mixed limit in the troposphere ( $T_{υ_{h}} \to T_{υ_{m}}$ ), a dry well-mixed limit in the stratosphere and mesosphere ( $T_{υ_{h}} \to T$ ), and a dry unmixed limit in the thermosphere ( $T_{υ_{h}} \to T_{υ_{t}}$ ).

View raw image

Fig. 3.

Profiles of specific gas constant R(z), temperature T(z), and thermospheric virtual temperature $T_{υ_{t}} = [R (z) / {\tilde{R}}_{d}] T (z)$ vs z, with the NAVGEM L160 upper boundary shown by the black solid line.

View raw image

Fig. 4.

Latitude–pressure plots of zonal-mean (a) thermospheric virtual temperature $T_{υ_{t}}$ (K) and (d) zonal wind U (m s⁻¹) as initialized on NAVGEM T119L160 grid. Corresponding plots below show zonal-mean errors in (b) $T_{υ_{t}}$ (K) and (e) U (10⁻² m s⁻¹) relative to these balanced initial states after 48 h of integration. (c),(f) The final zonal mean $T_{υ_{t}}$ and U errors after 120 h.

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Fig. 5.

Latitude–pressure plots of zonal-mean errors in +120-h forecasts of (a) temperature (K) and (c) zonal wind (10⁻² m s⁻¹), relative to the balanced initial state, using corrected HVPT $θ_{υ_{c}}$ with ς =1. (b),(d) The corresponding errors after 120 h when forecasts were rerun using ς = 10⁻⁹.

View raw image

Fig. 6.

The prescribed temperature fields on the NAVGEM T119L160 grid (a) at initialization, given by (73) with T_iso = 300 K, and (b) in the t → ∞ forecast limit, given by (73) with T_iso = 700 K (contour labels in K). For our α = 0.168 56 h⁻¹ Newtonian heating experiment, the thermosphere is heated to within 1% of (b) after ∼27.3 h.

View raw image

Fig. 7.

Zonal-mean (a) zonal wind and (d) meridional wind (uniformly zero) at 0 h using the balanced Ullrich et al. (2014) initial condition on the NAVGEM T119L160 grid capped by an isothermal 300-K thermosphere. Corresponding plots below show zonal-mean wind errors relative to these 0-h states as the thermosphere is heated to near 700 K: (b) zonal wind and (e) meridional wind errors after 48 h and (c) zonal wind and (f) meridional wind errors after 120 h.

View raw image

Fig. A1.

(left) Profiles of (a) R and (c) κ⁻¹, for F_10.7 of 100 sfu and A_p = 4 at the equator and Greenwich meridian on day 80, based on global-mean MSIS data for diurnally varying global-mean temperature and seven major thermospheric species, at six different local/universal times shown in panel key. Gold curves shows least squares fit of diurnal-mean profile over 0.1–10⁻⁸-hPa interval to the analytical functions (A1) and (A3). (right) Final fits to (b) $\hat{R} = R / {\tilde{R}}_{d}$ (blue) and (d) κ⁻¹ (maroon), using (A1) and (A3), respectively, with solid, dotted, and dashed curves showing fits for F_10.7 of 140, 80, and 200 sfu, respectively.

View raw image

Fig. A2.

(a) Solid curves show variation of a₀ (black), a₁ (blue), and a₂ (green) vs F_10.7 for $\tilde{R}$ profiles fitted to the function (A1), shown in 30 sfu steps between 50 and 290 sfu. Overlayed thicker transparent curves show third-order polynomial fits to these coefficient data, including behavior extrapolated outside the fitted data range. (b) Corresponding curves show the same data and fits for k₀, k₁, k₂, and k₃ coefficients of the κ⁻¹ profiles fitted to the function (A3).

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Editorial Type:: Article

Article Type:: Research Article

Adaptation of θ-Based Dynamical Cores for Extension into the Thermosphere Using a Hybrid Virtual Potential Temperature

Stephen D. Eckermann

Stephen D. Eckermann ^aSpace Science Division, U.S. Naval Research Laboratory, Washington, D.C.

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Cory A. Barton

Cory A. Barton ^aSpace Science Division, U.S. Naval Research Laboratory, Washington, D.C.

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James F. Kelly

James F. Kelly ^aSpace Science Division, U.S. Naval Research Laboratory, Washington, D.C.

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Online Publication:: 25 Jul 2023

Print Publication:: 01 Aug 2023

DOI:: https://doi.org/10.1175/MWR-D-22-0320.1

Page(s):: 1937–1955

Received:: 23 Nov 2022

Final Form:: 07 Apr 2023

Accepted:: 08 Apr 2023

Published Online:: 25 Jul 2023

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

The virtual temperature used to model moisture-modified tropospheric dynamics is generalized to include a new thermospheric component. The resulting hybrid virtual potential temperature (HVPT) transitions seamlessly with height, from moist virtual potential temperature (MVPT) in the troposphere, to potential temperature in the stratosphere and mesosphere, to thermospheric virtual potential temperature thereafter. For numerical weather prediction (NWP) models looking to extend into the thermosphere, but still heavily invested in retaining MVPT-based dynamical cores for tropospheric prediction, upgrading to HVPT allows the core to capture critical new aspects of variable composition thermospheric dynamics, while leaving the original MVPT-based tropospheric equations and numerics essentially untouched. In this way, HVPT augmentation can both simplify and streamline extension into the thermosphere at little computational cost beyond the inevitable need for more vertical layers and somewhat smaller time steps. To demonstrate, we upgrade the MVPT-based dynamical core of the Navy global NWP model to HVPT, then test its performance in forecasting analytical globally balanced states containing hot or rapidly heated thermospheres and height-varying gas constants. These tests confirm that HVPT augmentation offers an efficient and effective means of extending MVPT-based NWP models into the thermosphere to accelerate development of future ground-to-space NWP models supporting space weather applications. The related issues of variable gravitational acceleration and shallow-atmosphere approximations are also briefly discussed.

For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Stephen D. Eckermann, stephen.eckermann@nrl.navy.mil

Abstract

For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Stephen D. Eckermann, stephen.eckermann@nrl.navy.mil

Keywords: Atmosphere; Dynamics; General circulation models; Numerical weather prediction/forecasting; Primitive equations model; Space weather

1. Introduction

The composition of the atmosphere below ∼100-km altitude remains roughly constant due to turbulent mixing. This affords useful simplifications to the atmospheric fluid equations that can improve computational speed and efficiency of the discretized forms used as dynamical cores for operational numerical weather prediction (NWP).

As NWP models are progressively extended upward, these convenient lower atmospheric simplifications no longer apply (Akmaev 2011). Kinematic molecular viscosities increase approximately exponentially with height such that, above ∼100-km altitude, they become large enough to suppress turbulence. Absent turbulent mixing, thermospheric composition varies with height according to species mass. Photolysis also leads to atoms ultimately dominating molecules as the major mass component of the thermosphere. These changes cause the specific gas constant R, and the mass specific heats at constant pressure and volume, c_p and c_υ, respectively, to increase with height in the thermosphere and to vary in space and time in response to changes in species concentrations. Likewise, the monotonic decrease in gravitational acceleration g with height yields significantly smaller thermospheric values relative to the surface.

Modifications to a model’s dynamical core to allow these parameters to vary will typically apply at all altitudes. If extensive intrusive changes are involved, such modifications may present barriers to operational transition. For example, additional numerical complexity and overhead can jeopardize continued reliable delivery of tropospheric forecasts within inflexible time windows using the finite computational resources available at operational centers. Indeed, to forecast R, c_p, and c_υ as global time-varying state variables ideally requires new initialization, photochemistry, and transport code for a range of new thermospheric constituents. In addition to adding considerable new computational overhead, there are at present no operational observations of thermospheric constituents to assimilate as initial conditions and verification for these constituent forecasts, suggesting both questionable and largely unverifiable thermospheric skill impacts.

With these realities in mind, while also recognizing new needs to extend NWP models into the thermosphere to support space weather applications (e.g., Jackson et al. 2019; Berger et al. 2020), here we seek simpler and less intrusive dynamical core upgrades that have minimal impacts on how existing NWP dynamical cores are formulated and run in the lower atmosphere, yet capture salient aspects of variable composition thermospheric dynamics. While our development here focuses on the discretized hydrostatic primitive equations (HPEs) within the Navy Global Environmental Model (NAVGEM), as discussed in section 2, our hybrid virtual potential temperature concepts, developed in section 3, also apply to the many other atmospheric models using virtual potential temperature formulations, including nonhydrostatic models. Variable gravitational acceleration is briefly considered in section 4. The new dynamical core is tested and numerically verified in section 5 for two new idealized states representing hot thermospheres, one incorporating variable R and κ = R/c_p. Results are summarized in section 6.

2. NAVGEM governing equations

Since we will test the ideas to follow in NAVGEM, we introduce and develop the concepts to follow within the specific context of the NAVGEM discretized HPEs. We focus below on those variables that are treated as constants in the HPEs via approximations that are accurate in the lower atmosphere but inaccurate in the thermosphere: viz, R, c_p, c_υ, and g. Quantifying any two of the first three variables gives the third via Meyer’s relation:

c_{p} - c_{υ} = R .

(1)

The continuous forms of the NAVGEM HPEs are

\frac{D_{η} u}{D t} = u υ \frac{\tan φ}{a} + f υ - \frac{1}{a \cos φ} [{(\frac{\partial Φ}{\partial λ})}_{η} + {\tilde{c}}_{p_{d}} θ_{υ_{m}} {(\frac{\partial Π}{\partial p_{S}})}_{η} {(\frac{\partial p_{S}}{\partial λ})}_{η}] + F_{u},

(2a)

\frac{D_{η} υ}{D t} = - u^{2} \frac{\tan φ}{a} - f u - \frac{1}{a} [{(\frac{\partial Φ}{\partial φ})}_{η} + {\tilde{c}}_{p_{d}} θ_{υ_{m}} {(\frac{\partial Π}{\partial p_{S}})}_{η} {(\frac{\partial p_{S}}{\partial φ})}_{η}] + F_{υ},

(2b)

\frac{D_{η} θ_{υ_{m}}}{D t} = \frac{Q}{ρ {\tilde{c}}_{p_{d}} Π},

(2c)

\frac{\partial p_{S}}{\partial t} = - \int_{p_{top}}^{p_{S}} \nabla_{η} \cdot (v d p),

(2d)

comprising equations for zonal and meridional momentum balance, the first law of thermodynamics, and mass continuity, respectively, where

\frac{D_{η}}{D t} = {(\frac{\partial}{\partial t} + v \cdot \nabla)}_{η} + \dot{η} \frac{\partial}{\partial η}

(3)

is the material time derivative on a constant surface of an arbitrary vertical coordinate η (defined below). Here t is time, λ is longitude, φ is latitude, a is Earth’s radius, ∇ = a⁻¹[cos⁻¹φ(∂/∂λ), ∂/∂φ], f is the Coriolis parameter, v = (u, υ) is horizontal wind vector, (F_u, F_υ) are momentum forcing terms, Q is diabatic heating rate (expressed as time rate of change of energy density), p is pressure, p_S is surface pressure, ρ is density, Φ is geopotential,

θ_{υ_{m}} = T_{υ_{m}} Π^{- 1}

(4)

is the moist virtual potential temperature (MVPT),

T_{υ_{m}}

is moist virtual temperature,

Π = {(\frac{p}{p_{0}})}^{κ}

(5)

is the Exner function, p₀ is a constant reference-level pressure (typically 1000 hPa), and

κ = \frac{R}{c_{p}} .

(6)

As implemented in NAVGEM for lower atmospheric NWP, all terms in (6) are constants:

κ = {\tilde{κ}}_{d} = 2 / 7

R = {\tilde{R}}_{d} = 287.059 67 J {kg}^{- 1} K^{- 1}

, and

{\tilde{c}}_{p d} = {\tilde{R}}_{d} / {\tilde{κ}}_{d}

. The origins of these constant values and the relation of

T_{υ_{m}}

to air temperature T are explained in section 3 when variable composition is considered.

The HPEs are discretized on the sphere following Arakawa and Suarez (1983) and Hogan and Rosmond (1991), using a Lorenz grid (L-grid) in the vertical and a quadratic Gaussian latitude-longitude grid to allow horizontal gradient terms to be computed via spherical harmonic transforms. The HPEs are solved using semi-implicit semi-Lagrangian (SISL) methods (see Hogan et al. 2014). For η, NAVGEM adopts the implicit hybrid σ–p form (Simmons and Strüfing 1983)

p (η) = A (η) + B (η) [p_{S} - p_{top}]

(7)

that transitions layer pressures smoothly from terrain-following near the surface to isobaric above the tropopause (Eckermann 2009): A(η) and B(η) are the isobaric and terrain-following coefficients, respectively, and p_top is pressure at the model’s upper boundary. To avoid evaluating η explicitly, (3) is reformulated into a more convenient expression involving pressure derivatives on η surfaces, viz.

\dot{η} \frac{\partial}{\partial η} = E (η) {(\frac{\partial}{\partial p})}_{η},

(8a)

E (η) = {(\dot{η} \frac{\partial p}{\partial η})}_{η} = - B (η) \frac{\partial p_{S}}{\partial t} - \int_{p_{top}}^{p} \nabla_{η} \cdot (v d p) .

(8b)

a. Equations of state and hydrostatic balance

The equation of state is expressed as

ρ^{- 1} = α = \frac{{\tilde{R}}_{d} θ_{υ_{m}} Π}{p} = {\tilde{c}}_{p_{d}} θ_{υ_{m}} \frac{d Π}{d p},

(9)

where α is specific volume. The geopotential

Φ = \int_{0}^{z} g d z^{'} = g_{0} z,

(10)

where g₀ = 9.8066 m s⁻² is the constant value of g at the surface, z is geometric height, and

z = Φ / g_{0}

is geopotential height. Combining (10) with the equation of state (9) and the hydrostatic relation dp = −ρgdz yields

d Φ = gdz = - α d p = - {\tilde{c}}_{p_{d}} θ_{υ_{m}} d Π,

(11)

so that Φ is evaluated diagnostically as

Φ (η) = Φ_{s} + {\tilde{c}}_{p_{d}} \int_{Π (η)}^{Π (p_{S})} θ_{υ_{m}} d Π,

(12)

where

Φ_{s} = g_{0} h

(13)

is surface geopotential and h is terrain height.

b. Pressure gradient force

Since the horizontal pressure gradient force per unit mass,

f_{PGF} = (f_{PGF}^{λ}, f_{PGF}^{φ}) = - \nabla_{p} Φ,

(14)

involves horizontal gradients on an isobaric surface, we relate those to η-surface gradients using [e.g., (1-44) of Haltiner and Williams (1980)]

\nabla_{p} = \nabla_{η} + \frac{\partial}{\partial η} \nabla_{p} η,

(15)

so that the zonal and meridional components become, respectively,

f_{PGF}^{λ} = \frac{- 1}{a \cos φ} [{(\frac{\partial Φ}{\partial λ})}_{η} + {(\frac{\partial Φ}{\partial η})}_{λ} {(\frac{\partial η}{\partial λ})}_{p}],

(16a)

f_{PGF}^{φ} = \frac{- 1}{a} [{(\frac{\partial Φ}{\partial φ})}_{η} + {(\frac{\partial Φ}{\partial η})}_{φ} {(\frac{\partial η}{\partial φ})}_{p}] .

(16b)

The second terms within square parentheses in (16) are associated with terrain-following layers and can be re-expressed using chain-rule differentiation and the hydrostatic relation (11) as

f_{PGF}^{λ} = \frac{- 1}{a \cos φ} [(\frac{\partial Φ}{\partial λ}) + {\tilde{c}}_{p_{d}} θ_{υ_{m}} (\frac{\partial Π}{\partial p_{S}}) (\frac{\partial p_{S}}{\partial λ})],

(17a)

f_{PGF}^{φ} = \frac{- 1}{a} [(\frac{\partial Φ}{\partial φ}) + {\tilde{c}}_{p_{d}} θ_{υ_{m}} (\frac{\partial Π}{\partial p_{S}}) (\frac{\partial p_{S}}{\partial φ})],

(17b)

with all derivatives in (17) evaluated on η surfaces, providing the final η-coordinate forms for pressure gradient force in the NAVGEM momentum equations, Eqs. (2a) and (2b).

3. Variable composition and new virtual potential temperature forms

a. Variable gas constants

For an atmosphere comprised of i = 1, …, I_tot mass constituents with individual mass mixing ratios r_i and specific gas constants

{\tilde{R}}_{i}

, the bulk specific gas constant

R = \sum_{i = 1}^{I_{tot}} r_{i} {\tilde{R}}_{i} .

(18)

Similarly, c_p is evaluated using ideal-gas relations as a mass-mixing-ratio-weighted sum of the individual species terms:

c_{p} = \sum_{i = 1}^{I_{tot}} r_{i} c_{p_{i}} = k_{B} \sum_{i = 1}^{I_{tot}} \frac{(1 + \frac{β_{i}}{2}) r_{i}}{m_{i}},

(19)

where β_i is the total number of degrees of freedom of gas species i, m_i is its mass, and k_B is the Boltzmann constant. Defining a species volume mixing ratio q_i and noting that

\sum_{i} q_{i} = 1

, we can rearrange (19) into an equivalent expression that scales the precomputed bulk specific gas constant R, viz.

c_{p} = R \sum_{i = 1}^{I_{tot}} (1 + \frac{β_{i}}{2}) q_{i},

(20a)

= R κ^{- 1} .

(20b)

Diatomic molecules have three translational and two rotational degrees of freedom (β_i = 5). Since ∼99% of the atmosphere from 0 to ∼100 km comprises the diatomic molecules N₂ and O₂, then the sum in (20a) is approximately constant:

κ^{- 1} = {\tilde{κ_{d}}}^{- 1} = {\tilde{c}}_{p_{d}} / {\tilde{R}}_{d} = 7 / 2

, explaining the values of these constants presented earlier in section 2.

Solid curves in Fig. 1a show representative profiles of R, κ⁻¹, and c_p as a function of the solar 10.7 cm solar radio flux (F_10.7) index for atmospheric pressures extending from the surface (1000 hPa) to near the exobase at ∼500 km (10⁻⁸ hPa), based on calculations and parameterized fits described in the appendix. We see in Fig. 1a that R increases by up to 100% in the thermosphere relative to its constant lower atmospheric value. Corresponding changes in c_p in Fig. 1c are lower, due to the accompanying decrease in κ⁻¹ with height in Fig. 1b.

Fig. 1.

Citation: Monthly Weather Review 151, 8; 10.1175/MWR-D-22-0320.1

While (18)–(20b) are bulk terms involving all mass constituents, in what follows we will split and separate the tropospheric and thermospheric contributions, since they are widely separated in altitude and thus independent of one another. This separation also helps to illustrate how the new thermospheric contributions to virtual temperature relate to the original moist tropospheric terms.

b. Moist virtual potential temperature (MVPT)

Using the equation of state to equate total pressure and partial pressures of dry air and water vapor, having densities ρ_d and ρ_w, respectively, yields

p = (ρ_{d} + ρ_{w}) R_{m} T = ({\tilde{R}}_{d} ρ_{d} + {\tilde{R}}_{w} ρ_{w}) T

(21)

and the moisture-modified bulk specific gas constant

R_{m} = {\tilde{R}}_{d} [1 + (\frac{{\tilde{R}}_{w}}{{\tilde{R}}_{d}} - 1) s_{w}] \approx (1 + 0.608 s_{w}) {\tilde{R}}_{d},

(22)

where

{\tilde{R}}_{w} = 461.52 J {kg}^{- 1} K^{- 1}

s_{w} = \frac{r_{w}}{1 + r_{w}}

(23)

is specific humidity, and r_w is the mass mixing ratio of water vapor. Since R_m can differ from the constant dry-air value

{\tilde{R}}_{d}

, yet constant

R = {\tilde{R}}_{d}

is hardwired into the dynamical core, then transforming to a moist virtual temperature:

T_{υ_{m}} \equiv \frac{R_{m}}{{\tilde{R}}_{d}} T = T [1 + (\frac{{\tilde{R}}_{w}}{{\tilde{R}}_{d}} - 1) s_{w}] \approx T (1 + 0.608 s_{w})

(24)

gives the exact pressure p (and hence exact pressure gradient forces) via the equation of state:

p = ρ {\tilde{R}}_{d} T_{υ_{m}} \equiv ρ R_{m} T .

(25)

The key point here is that by virtualizing our prognostic potential temperature variable $θ_{υ_{m}}$ in the HPEs (2) to MVPT via (4) and (24), the dynamical core can now capture aspects of moisture-modified dynamics due to moist changes in total mass composition, and hence in the bulk specific gas constant R, despite retaining a constant “dry” value of $R = {\tilde{R}}_{d}$ within the discretized HPEs. This approach is of course not unique to NAVGEM but is incorporated into the dynamical cores of many other NWP models (e.g., Klemp and Wilhelmson 1978; Hodur 1997; Davies et al. 2005; Skamarock et al. 2012, 2021; Walters et al. 2017; Polichtchouk et al. 2020).

The geopotential (12) can now be re-expressed using (9) and (24) as

Φ (η) = Φ_{s} + {\tilde{R}}_{d} \int_{p (η)}^{p_{S}} T_{υ_{m}} d \log p,

(26)

= Φ_{s} + \int_{p (η)}^{p_{S}} (R_{m} T) d \log p,

(27)

with (27) illustrating that moist virtual temperature incorporates an exact moisture-modified gas constant R_m within the geopotential pressure integral, yielding exact moisture-modified geopotentials, and hence exact pressure gradient forces −∇_pΦ via (17) in the momentum equations.

While for simplicity we have considered only the gas phase here, it is straightforward to include additional moist modifications due to liquid water and ice (e.g., Davies et al. 2005).

c. Thermospheric virtual potential temperature

To incorporate changes in R due to variations in the composition of the thermosphere, we can define an analogous thermospheric virtual temperature,

T_{υ_{t}} = T \frac{R_{t}}{{\tilde{R}}_{d}},

(28)

where R_t is a thermospheric specific gas constant calculated using (18) assuming no moisture contribution (s_w = 0). This derivation further assumes that changes in thermospheric species concentrations are primarily due to neutral photochemistry and transport that lead to no net loss of total mass. This is not strictly true since some thermospheric neutrals are photoionized during the day, but these ions and electrons are trace constituents and largely recombine into neutrals at night, and even as ions arguably still contribute to the total space–time distribution of species concentrations controlling bulk specific gas constants.

While we only show vertical profiles of thermospheric R_t in Fig. 1a, we note that the virtual thermospheric temperature (28) and the hybrid forms that follow all hold for an R_t that can vary in all three spatial dimensions and in time, just like s_w in (24).

d. Hybrid virtual potential temperature (HVPT)

Next, we combine these separate moist and thermospheric composition-induced changes into a bulk specific gas constant for the entire atmosphere, which yields the hybrid virtual temperature:

T_{υ_{h}} = T [1 + ({\hat{R}}_{t} - 1) + ({\hat{\tilde{R}}}_{w} - 1) s_{w} + (1 - {\hat{R}}_{t}) s_{w}] .

(29)

Here, hatted Rs are normalized by the constant lower atmospheric value

{\tilde{R}}_{d}

, all tilded Rs are constants, whereas untilded Rs vary in space and time, viz.

{\hat{R}}_{t} = \frac{R_{t}}{{\tilde{R}}_{d}},

(30a)

{\hat{\tilde{R}}}_{w} = \frac{{\tilde{R}}_{w}}{{\tilde{R}}_{d}} \approx 1.608 .

(30b)

The second and third terms inside square brackets in (29) are the modification terms for thermospheric and moist composition changes, respectively, presented earlier. The fourth is an interaction term involving the product of thermospheric and moist composition changes, which modifies the

{\hat{\tilde{R}}}_{w} = 1.608

in (29) to account for the change in thermospheric R_t away from

{\tilde{R}}_{d}

: in practice this term vanishes since the thermosphere and moist troposphere are widely separated in altitude.

In modifying our vertically extended dynamical core to use a hybrid virtual potential temperature (HVPT),

θ_{υ_{h}} = T_{υ_{h}} Π^{- 1},

(31)

the equation sets in section 2 remain unchanged except that the moist form

θ_{υ_{m}}

is now replaced everywhere by

θ_{υ_{h}}

. Figure 2 shows schematically how hybrid virtual temperature

T_{υ_{h}}

seamlessly reproduces limiting virtual temperature forms within various layers of the atmosphere. Specifically, it asymptotically approaches the moist virtual temperature

T_{υ_{m}}

in the troposphere, the dry air temperature T in the stratosphere and mesosphere, and the thermospheric virtual temperature

T_{υ_{t}}

in the thermosphere.

Fig. 2.

Citation: Monthly Weather Review 151, 8; 10.1175/MWR-D-22-0320.1

Equivalent forms of (29) are

T_{υ_{h}} = T [{\hat{R}}_{t} + ({\hat{\tilde{R}}}_{w} - {\hat{R}}_{t}) s_{w}],

(32a)

= T_{υ_{m}} + T ({\hat{R}}_{t} - 1) (1 - s_{w}),

(32b)

= T_{υ_{m}} [1 + \frac{({\hat{R}}_{t} - 1) (1 - s_{w})}{1 + ({\hat{\tilde{R}}}_{w} - 1) s_{w}}] .

(32c)

Note again how (32a) reproduces

T_{υ_{m}}

in (24) as

{\hat{R}}_{t} \to 1

and

T_{υ_{t}}

in (28) as s_w → 0. Next (32a) is re-expressed in (32b) as an additive modification of the standard moist virtual temperature

T_{υ_{m}}

, which is essentially zero below ∼100 km where

{\hat{R}}_{t} \to 1

. In (32c)

T_{υ_{m}}

is modified via a multiplicative scaling factor, which is close to unity below ∼100 km. Equation (32c), and the corresponding inverse transformation

T_{υ_{m}} = T_{υ_{h}} [1 - \frac{({\hat{R}}_{t} - 1) (1 - s_{w})}{{\hat{R}}_{t} + ({\hat{\tilde{R}}}_{w} - {\hat{R}}_{t}) s_{w}}],

(33)

prove to be useful forms for use in NAVGEM, since the MVPT,

θ_{υ_{m}}

, has been used for years in the evolving U.S. Navy global NWP models (Hogan and Rosmond 1991), such that various legacy tropospheric physics, initialization, data assimilation and postprocessing codes require MVPT as inputs. Since the corresponding hybrid form

θ_{υ_{h}}

is needed for dynamics updates in new high-altitude configurations extending into the thermosphere, (32c) and (33) allow us to move seamlessly back and forth between

θ_{υ_{h}}

and

θ_{υ_{m}}

as needed within different parts of the code.

e. Mass specific heats using HVPT $θ_{υ_{h}}$

Differentiating the ideal-gas equation

p α = R T = {\tilde{R}}_{d} T_{υ_{h}}

yields

\frac{D (R T)}{D t} = {\tilde{R}}_{d} \frac{D T_{υ_{h}}}{D t} = p \frac{D α}{D t} + α \frac{D p}{D t},

(34)

which can be expressed equivalently using Meyer’s relation (1) as

\frac{D [(c_{p} - c_{υ}) T]}{D t} = ({\tilde{c}}_{p_{d}} - {\tilde{c}}_{υ_{d}}) \frac{D T_{υ_{h}}}{D t} = p \frac{D α}{D t} + α \frac{D p}{D t} .

(35)

This equation comprises the difference between prognostic equations for specific internal energy,

\frac{D (c_{υ} T)}{D t} + p \frac{D α}{D t} = {\tilde{c}}_{υ_{d}} \frac{D T_{υ_{h}}}{D t} + p \frac{D α}{D t} = ρ^{- 1} Q,

(36)

and specific enthalpy,

\frac{D (c_{p} T)}{D t} - α \frac{D p}{D t} = {\tilde{c}}_{p_{d}} \frac{D T_{υ_{h}}}{D t} - α \frac{D p}{D t} = ρ^{- 1} Q .

(37)

Inspecting (36) and (37) immediately reveals that c_υ and c_p variabilities are included implicitly through

T_{υ_{h}}

via the explicit diatomic molecular forms

c_{υ} = R ({\tilde{κ}}_{d}^{- 1} - 1) = (5 / 2) R

and

c_{p} = R {\tilde{κ}}_{d}^{- 1} = (7 / 2) R

. In short, temperature virtualization to

T_{υ_{h}}

restricts to constant κ, which in turn implies variable c_p and c_υ since R now varies. For each R profile in Fig. 1a, dotted curves in Fig. 1c show the corresponding

c_{p} = {\tilde{κ}}_{d}^{- 1} R

profile that results from constant

{\tilde{κ}}_{d}^{- 1} = 7 / 2

in the HVPT dynamical core. In the thermosphere these profiles are all systematically larger than the actual c_p profiles in Fig. 1c since, as shown in Fig. 1b, κ⁻¹ decreases with height.

The above expressions and results are fully consistent with our prognostic

θ_{υ_{h}}

formulation. The middle expression in (37) can be re-expressed using the equation of state as

\frac{D \log T_{υ_{h}}}{D t} - {\tilde{κ}}_{d} \frac{D \log p}{D t} = \frac{Q}{ρ T_{υ_{h}} {\tilde{c}}_{p_{d}}} .

(38)

Using the HVPT as defined in (31), the left-hand side of (38) is equal to

D \log θ_{υ_{h}} / D t

, which, when inserted into (38), yields our HVPT-based thermodynamic equation

\frac{D θ_{υ_{h}}}{D t} = \frac{Q}{ρ {\tilde{c}}_{p_{d}} Π}

(39)

that augments the standard prognostic MVPT equation in the HPEs, given by Eq. (2c).

f. Variable κ: A corrected HVPT $θ_{υ_{c}}$

Next, we explore building variable κ into a θ-based dynamical core using a corrected HVPT variable.

1) Specific enthalpy

For variable κ atmospheres, Akmaev and Juang (2008) advocated changing the prognostic thermodynamic variable from potential temperature to specific enthalpy (either its absolute or potential form). This is problematic since the specific enthalpy equation, Eq. (37), involves the material time derivative of p, which generally requires solving a separate tendency equation for pressure. For the special case of the HPEs, Dp/Dt is diagnostic rather than prognostic, given as

ω = \frac{D_{η} p}{D t} = B (η) [v \cdot \nabla_{η} p_{S}] - \int_{p_{top}}^{p (η)} \nabla_{η} \cdot (v d p) .

(40)

In reference to temperature virtualization, Akmaev and Juang (2008) cite a modified form of the specific enthalpy equation used in the European Centre for Medium-Range Weather Forecasts (ECMWF) SISL NWP model that solves for temperature, viz. (Ritchie et al. 1995)

\frac{D T}{D t} - T_{υ_{h}} (\frac{κ ω}{p}) = \frac{Q}{ρ c_{p}} .

(41)

Akmaev and Juang (2008) argue that the mixing of absolute temperature T and virtual temperature

T_{υ_{h}}

in this equation is evidence of the questionable usefulness of virtual temperature as a prognostic thermodynamic variable in models, since reformulating (41) into a prognostic equation for

T_{υ_{h}}

requires additional problematic “correction” terms.

Such arguments misinterpret (41) as an exact form. Differentiating the exact specific enthalpy equation, given by the expressions on the left- and right-hand sides of (37) and assuming exact (rather than HVPT-approximated) c_p, gives the pure temperature form

\frac{D T}{D t} - T (\frac{κ ω}{p}) + T (c_{p}^{- 1} \frac{D c_{p}}{D t}) = \frac{Q}{ρ c_{p}},

(42)

where κ, R, and c_p are all exact. This equation can now be rearranged into two equivalent forms: a mixed

T – T_{υ_{h}}

form

\frac{D T}{D t} - T_{υ_{h}} (\frac{κ_{*} ω}{p}) + T (c_{p}^{- 1} \frac{D c_{p}}{D t}) = \frac{Q}{ρ c_{p}},

(43)

for comparison with (41), where

κ_{*} = {\tilde{R}}_{d} / c_{p}

, and a pure virtualized

T_{υ_{h}}

form

\frac{D T_{υ_{h}}}{D t} - T_{υ_{h}} (\frac{κ ω}{p}) - T_{υ_{h}} (κ^{- 1} \frac{D κ}{D t}) = \frac{Q κ}{ρ {\tilde{R}}_{d}} .

(44)

Comparing (41) and (43) reveals that (41) is an approximate form that assumes c_p is locally constant (Dc_p/Dt = 0), whereupon

κ_{*}

is also locally constant. Comparing these equations to (44) shows that an analogous form of (41) involving only

T_{υ_{h}}

follows by assuming κ is locally constant (Dκ/Dt = 0). Thus, the supposed shortcomings of using

T_{υ_{h}}

rather than T within these approximated “locally constant” temperature-tendency forms of the specific enthalpy equation do not exist. Both the mixed form (43) and the fully virtualized form (44) contain additional material derivatives that require an approximation that these material derivative terms are small enough to omit. Of course (42)–(44) indicate that one is free here to choose either T or

T_{υ_{h}}

as the preferred prognostic variable. Note, however, that despite solving for T, (41) and (43) still insert a virtualized temperature variable into these equations to play the same pivotal role in converting the constant

{\tilde{R}}_{d}

within the dynamical core into its needed variable form R in these equations, and virtual temperature is also retained in the momentum equations and geopotential relations to produce exact pressure gradient forces in the momentum equations (see, e.g., section 2a of Ritchie et al. 1995). Ritchie et al. (1995) also discuss efficiencies of using

T_{υ_{h}}

rather than T to reduce the number of spectral transforms required for the semi-Lagrangian (SL) calculations.

Note, however, that we can now define a corrected hybrid virtual temperature

T_{υ_{c}} = {(\frac{κ}{{\tilde{κ}}_{d}})}^{- 1} T_{υ_{h}} \equiv (\frac{c_{p}}{{\tilde{c}}_{p_{d}}}) T

(45)

such that the exact virtualized specific enthalpy equation given by Eq. (44) becomes

\frac{D T_{υ_{c}}}{D t} - T_{υ_{c}} (\frac{κ ω}{p}) = \frac{Q}{ρ {\tilde{c}}_{p_{d}}} .

(46)

The problematic Dκ/Dt term in (44) is now entirely removed from this exact form of the specific enthalpy equation without resorting to a locally constant approximation of Dκ/Dt ≈ 0, and without any requirement to evaluate Dκ/Dt (or Dc_p/Dt) explicitly, only κ.

2) Corrected HVPT

By analogy to

T_{υ_{c}}

in (45) and (46), a corrected HVPT can now be defined as

θ_{υ_{c}} = T_{υ_{c}} {(\frac{p}{ς p_{0}})}^{- κ} = θ_{υ_{h}} ς^{{\tilde{κ}}_{d}} {(\frac{κ}{{\tilde{κ}}_{d}})}^{- 1} {(\frac{p}{ς p_{0}})}^{{\tilde{κ}}_{d} - κ} = θ_{υ_{h}} Π_{υ_{c}}^{- 1},

(47)

where

Π_{υ_{c}} = ς^{- {\tilde{κ}}_{d}} (\frac{κ}{{\tilde{κ}}_{d}}) {(\frac{p}{ς p_{0}})}^{κ - {\tilde{κ}}_{d}}

(48)

is an Exner correction that relates

θ_{υ_{h}}

and

θ_{υ_{c}}

via (47). The constant scaling factor ς is set to unity for now.

The major change to the virtual potential temperature definition in (47) is the ${(κ / {\tilde{κ}}_{d})}^{- 1}$ factor, which generalizes virtual potential temperature to capture the $T_{υ_{h}}$ (D logκ/Dt) tendency term in the virtualized specific enthalpy equation given by Eq. (44). Note that this new term becomes unity below ∼100 km where $κ \to {\tilde{κ}}_{d}$ , thus maintaining the standard MVPT form of the equations in the lower atmosphere.

With this general definition, the prognostic equation

\frac{D θ_{υ_{c}}}{D t} = \frac{Q}{ρ {\tilde{c}}_{p_{d}} Π Π_{υ_{c}}} + θ_{υ_{c}} \frac{D κ}{D t} [\log ς - \log (\frac{p}{p_{0}})]

(49)

reproduces the exact form of the virtualized specific enthalpy equation given by Eq. (46).

The corresponding

θ_{υ_{h}}

equation for variable κ is

\frac{D θ_{υ_{h}}}{D t} + θ_{υ_{h}} (κ - {\tilde{κ}}_{d}) (\frac{ω}{p}) = \frac{Q}{ρ {\tilde{c}}_{p_{d}} Π} + θ_{υ_{h}} κ^{- 1} \frac{D κ}{D t} .

(50)

In this case, variable κ adds a new vertical advection term as in the prognostic temperature Eqs. (42)–(44), which destroys the compact material-derivative form that is the primary advantage of using potential temperature in the dynamical core. By contrast, using

θ_{υ_{c}}

yields a prognostic equation given by Eq. (49) that retains the compact material-derivative form used within standard

θ_{υ_{m}}

-based NWP cores, and so we focus on it as less intrusive potential augmentation to an existing NWP dynamical core.

A new Dκ/Dt term appears in (49), needed to remove an identical term contained within the material derivative of $θ_{υ_{c}}$ that is absent from the specific enthalpy equation, given by Eq. (46). For the standard potential temperature with ς = 1, the logς term in square parentheses in (49) vanishes, leaving a −log(p/p₀) correction term that has been noted previously for prognostic equations for dry potential temperature (e.g., Klemp and Skamarock 2021) and potential enthalpy (Juang 2011) when κ varies.

The appearance of Dκ/Dt terms in (49) is unavoidable since potential temperature is no longer conserved in variable-κ atmospheres. Evaluating Dκ/Dt generally requires solutions from additional prognostic equations governing constituent transport and photochemistry (e.g., Juang 2011; Liu et al. 2018), which are nontrivial and potentially expensive additions. For our one-dimensional profile on isobaric η surfaces in Fig. 1b, Dκ/Dt simplifies to ω(∂κ/∂p), where ∂κ/∂p is given by the analytical p derivative of the fitted expression (A3) (if F_10.7 varies during the forecast, a ∂κ/∂t term is also required).

So, the question now arises as to whether the magnitude of the Dκ/Dt term in (49) can be minimized to make a locally constant approximation of Dκ/Dt ≈ 0 more tenable. Using the standard p₀ = 1000 hPa, the −log(p/p₀) correction term in (49) becomes large and positive in the thermosphere, since p/p₀ ≪ 1. The ς factor in (48) allows us to choose a modified

{\overset{⌣}{p}}_{0} = ς p_{0}

(51)

for deep thermospheric applications that modifies this term to

- \log (p / {\overset{⌣}{p}}_{0})

and reduces its magnitude. For example, setting ς = 10⁻⁹ yields

{\overset{⌣}{p}}_{0}

= 10⁻⁶ hPa, a nominal pressure level where κ is undergoing greatest vertical variations in Fig. 1b. At p =

{\overset{⌣}{p}}_{0}

= 10⁻⁶ hPa, the Dκ/Dt term in (49) vanishes and the solution is exact, while at pressures a decade above and below

{\overset{⌣}{p}}_{0}

, the magnitude of this term is an order of magnitude smaller than when using

{\overset{⌣}{p}}_{0}

= p₀ = 1000 hPa (ς = 1). While these changes make

| \log (p / {\overset{⌣}{p}}_{0}) |

large at lower altitudes, here

κ \to {\tilde{κ}}_{d}

and so the Dκ/Dt term in (49) vanishes anyway below ∼10⁻⁴ hPa (cf. Fig. 1b). While

θ_{υ_{c}}

values are also smaller when using ς = 10⁻⁹, note from (49) that these estimated error reductions apply to

θ_{υ_{c}}^{- 1} (D θ_{υ_{c}} / D t)

for a given value of Dκ/Dt, implying smaller relative errors in

θ_{υ_{c}}

irrespective of its absolute value.

With these changes, setting Dκ/Dt = 0 as a more tenable approximation in (49) gives the compact equation:

\frac{D θ_{υ_{c}}}{D t} = \frac{Q}{ρ {\tilde{c}}_{p_{d}} Π Π_{υ_{c}}} .

(52)

When using

θ_{υ_{c}}

, the constant-κ geopotential relation (12) must also be corrected to reproduce the exact κ-independent relation (26). For the Arakawa and Suarez (1983) discretization the modification is

Φ (η) = Φ_{s} + {\tilde{c}}_{p_{d}} \int_{Π (η)}^{Π (p_{S})} θ_{υ_{c}} Π_{υ_{c}} d Π,

(53)

where the integral retains the constant-

{\tilde{κ}}_{d}

form of the Exner function (5), since it is used on all model levels for the vertical interpolation of pressure and state variables. Likewise, the

θ_{υ_{m}}

terms in (17) are replaced by

θ_{υ_{c}} Π_{υ_{c}}

, but note that these terms vanish within isobaric model layers in the stratosphere and above when using a hybrid vertical coordinate (7), since ∂Π/∂p_S = 0, in which case (17) could continue to be used since these expressions remain exact on terrain-following tropospheric layers, since

κ = {\tilde{κ}}_{d}, R = {\tilde{R}}_{d}

and thus

Π_{υ_{c}} \to 1

and

θ_{υ_{c}} \to θ_{υ_{m}}

3) Corrected potential temperature and potential enthalpy

It proves useful at this point to consider briefly a closely related class of NWP models that use potential temperature θ = TΠ⁻¹ as the prognostic variable, then derive $θ_{υ_{m}}$ diagnostically from θ to evaluate the equation of state, pressure-gradient forces in the momentum equations and, for hydrostatic models, diagnostic geopotentials (e.g., Davies et al. 2005; Polichtchouk et al. 2020). In such models it is also straightforward to generalize to standard HVPT by augmenting the diagnostic step of converting θ to $θ_{υ_{h}}$ instead of $θ_{υ_{m}}$ . Such models incorporate a constant thermospheric $c_{p} = {\tilde{c}}_{p_{d}}$ into the core’s adiabatic temperature solutions, whereas our approach of using standard HVPT as the prognostic variable incorporates height-varying $c_{p} = R / {\tilde{κ}}_{d}$ (dotted curves in Fig. 1c) into the adiabatic temperature solutions.

As evident in Fig. 1c, neither approach incorporates realistic thermospheric c_p variability and so both require a corrected HVPT for more accurate adiabatic temperature solutions. Incorporating corrected HVPT into models that predict θ requires migrating to a corrected potential temperature

θ_{c} = θ Π_{c}^{- 1}

(54a)

via the θ-specific Exner correction

Π_{c} = ς^{- {\tilde{κ}}_{d}} {(\frac{c_{p}}{{\tilde{c}}_{p_{d}}})}^{- 1} {(\frac{p}{ς p_{0}})}^{κ - {\tilde{κ}}_{d}},

(54b)

then solving a prognostic θ_c equation of either the general form (49) or approximated form (52), such that θ_c and Π_c now replace

θ_{υ_{c}}

and

Π_{υ_{c}}

in these equations. The c_p dependence in (54b) results from the conversion of T to

T_{υ_{c}}

in (45), such that prognostic θ_c now provides an indirect solution to the potential enthalpy equation (cf. Juang 2011). Prognostic θ_c is then converted diagnostically into standard HVPT

θ_{υ_{h}}

as before for use in the momentum equations, equation of state, and for computing hydrostatic geopotentials. Since

θ_{c} \equiv θ_{υ_{c}}, θ_{υ_{c}}

too can be viewed as further augmenting the virtualization of potential temperature within a θ-based dynamical core to provide an indirect potential enthalpy solution that improves the prediction of variable-c_p thermospheres.

4) Standard HVPT errors due to constant $κ = {\tilde{κ}}_{d}$

What other typical errors do we anticipate when using a globally constant $κ = {\tilde{κ}}_{d}$ in a standard (uncorrected) HVPT dynamical core?

Errors in κ affect the way parcels expand and contract, and hence cool or heat with respect to the local background temperature, under adiabatic vertical transport. They will clearly be greatest when the thermosphere undergoes significant vertical upwelling or downwelling. Typically, however, most thermospheric vertical advection is driven by gravity waves and tides, and is periodic and reversible in the absence of strong dissipation. Consider the isobaric form of buoyancy frequency, which, for locally constant R_t and κ, is given by

N = {\frac{R_{t}}{H_{ref}} [\frac{\partial T}{\partial Z} + \frac{κ T}{H_{ref}}]}^{1 / 2},

(55)

where

Z = H_{ref} \log (\frac{p_{0}}{p})

(56)

is pressure height and H_ref is a reference pressure scale height (typically set to 7 km in the lower atmosphere). Buoyancy frequencies affect both the frequency and wave-induced temperature amplitudes of resolved gravity waves. Likewise, the speed of sound

C_{s} = {(\frac{R_{t} T}{1 - κ})}^{1 / 2} = {(\frac{{\tilde{R}}_{d} T_{υ_{t}}}{1 - κ})}^{1 / 2},

(57)

broadly characterizes the fastest horizontal phase speed modes (Lamb modes and gravest vertical-mode gravity waves) for the semi-implicit time integration (Davies et al. 2003).

The specific κ dependences in (55) and (57) reduce the impacts of thermospheric κ biases on these key wave parameters. For example, reading off κ values from Fig. 1b, the high bias in C_s from using constant ${(1 - {\tilde{κ}}_{d})}^{- 1 / 2}$ in (57) is ∼3.5% at 10⁻⁶ hPa and ∼8% at 10⁻⁸ hPa. These offsets seem tolerably small, given that semi-implicit time integration artificially slows down phase speeds of the fastest wave modes anyway (Simmons and Temperton 1997).

The effect of κ bias on N is diminished by noting that the term in square parentheses in (55) is dominated by ∂T/∂Z, which is large and positive in the thermosphere, with the κ-dependent lapse rate much smaller in magnitude by comparison. Errors become larger in the upper thermosphere, where ∂T/∂Z → 0 and κ departures from ${\tilde{κ}}_{d}$ are largest (see Fig. 1b). Thus corrected HVPT offers potentially significant error reductions for models extending to very high altitudes.

Compare these errors to those from omitting temperature virtualization and assuming constant R, which scale as R^1/2 for both N and C_s. Reading off R values from Fig. 1a, they are ∼21% at 10⁻⁶ hPa and ∼36% at 10⁻⁸ hPa, about 4–6 times higher than the corresponding relative errors in C_s from assuming constant κ. These errors are in addition to the huge errors in geopotential and pressure gradient force that result from using constant R in the thermosphere, as previously discussed.

Thus, we conclude that an upgraded HVPT-based dynamical core represents an adequate bridging strategy to allow an existing dynamical core to be extended into the thermosphere to capture realistic composition-induced changes to the equations of state, hydrostatic balance, and momentum balance, with acceptable thermodynamic error biases due to use of constant κ. In cases where accurate κ distributions are required in the forecast, such as deeper vertical extensions into the thermosphere, the corrected HVPT form (49) or its locally constant approximation (52) can be employed.

g. Diabatic tendency Q

We now briefly consider the right-hand sides of the HVPT Eqs. (39) and (49), where diabatic heating due to physics is represented as a net energy tendency Q. Since physics terms are more naturally applied in NAVGEM as temperature tendencies, then after the dynamics step and immediately before the physics update, we convert HVPT (either $θ_{υ_{h}}$ or $θ_{υ_{c}}$ ) to temperature T. Then, consistent with the equivalent temperature form (42) of the HVPT equation, we apply physics tendencies Q/ρc_p to update T cumulatively, where c_p is the exact profile computed from (19). As discussed earlier and as can be shown using (37), when using $θ_{υ_{h}}$ rather than $θ_{υ_{c}}$ , c_p in (42) is replaced everywhere by the uncorrected HVPT form of c_p given as $R {\tilde{κ}}_{d}^{- 1}$ . In this case too we use the exact c_p for the tendency Q/ρc_p to ensure a proper diabatic heat budget in the thermosphere. Once completed, we convert temperature back to HVPT prior to the next dynamics step.

The NWP community has revisited thermodynamic potentials as a more fundamental basis for self-consistently deriving thermodynamic equations that capture latent heat terms due to phase changes in water within the dynamical core (Thuburn 2017; Staniforth and White 2019). In this work we have assumed for simplicity that latent heating rates are encapsulated within Q. Composition changes due to ion-neutral photochemistry can also change specific thermodynamic potentials such as enthalpy [see section 2.2 of Eldred et al. (2022)]. Again, for simplicity we have made no attempt here to extract any additional composition-dependent chemical potential terms for explicit inclusion in the specific enthalpy or HVPT equations, assuming instead that any heating due to thermospheric composition changes is also encapsulated within Q. For example, heating of the mesosphere and thermosphere due to exothermic photochemical reactions that alter composition are typically parameterized as a direct diabatic contribution to Q (e.g., Marsh et al. 2007). Despite these differences, our investigation of errors due to a constant-κ prognostic virtual potential temperature variable has initiated a similarly motivated preliminary error assessment for the thermosphere to those deeper tropospheric assessments based on thermodynamic potentials [cf. sections 3 and 4 of Eldred et al. (2022)].

4. Variable gravitational acceleration

We now briefly consider constant versus variable g. Use of constant g is generally understood to follow from the shallow-atmosphere approximation used to derive the HPEs (White et al. 2005). In the lower atmosphere the errors from such an approximation are self-evidently small. For example, scale analyses suggest other terms removed by the shallow-atmosphere approximation, such as cosφ Coriolis terms, yield greater errors by comparison (White and Bromley 1995).

This is no longer the case in the thermosphere, where

g (z) = \frac{g_{0}}{{(1 + z / a)}^{2}}

(58)

is ∼10% lower than at the surface. Deng et al. (2008) investigated how simulated thermospheric dynamics varied within their deep-atmosphere nonhydrostatic thermospheric model using variable and constant g, finding appreciable differences that suggested shortcomings in applying the constant-g HPEs in the thermosphere. They also stated (without proof) that incorporation of variable g into HPEs did not appear to be straightforward, since doing so could lead to imbalances between mass and wind fields. Though rarely discussed, the vast majority of whole-atmosphere models (WAMs) based on deep vertical extensions of dynamical cores using HPEs appear to retain constant g: for some explicit discussion of this issue, see Maute (2017) and Liu et al. (2018).

In his influential review paper on whole-atmosphere modeling, Akmaev (2011) challenged the need for constant-g within ground-to-thermosphere models using the HPEs. His arguments focus on the unusual property of the HPEs, stemming from their use of pressure as the vertical coordinate, that g disappears from the discretized equations, being subsumed via hydrostatics into a temperature-dependent Φ given in (12). Akmaev (2011) argued that the absence of g in the HPEs implies that their prognostic dynamics are (quote) “formally independent of whether or not g is constant.” Paraphrasing his arguments, the HPEs provide solutions on model pressure surfaces that are valid for any choice of g(z), subject only to the Φ_S lower boundary condition in (12). Relatedly, Akmaev (2011) also rejected arguments that variable g violates the shallow-atmosphere approximation (see, e.g., section 3b of White et al. 2005), arguing that the choice for g depends only on hydrostatics that exist largely separate from and independent of the shallow-atmosphere approximations.

For any vertical coordinate η, mass continuity takes the general form [Eq. (3.21) of Kasahara 1974]:

\frac{\partial}{\partial t} {(ρ \frac{\partial z}{\partial η})}_{η} + \nabla_{η} \cdot (v ρ \frac{\partial z}{\partial η}) + \frac{\partial}{\partial η} (\dot{η} ρ \frac{\partial z}{\partial η}) = 0,

(59)

where v represents velocity along a constant-η surface. For our specific hybrid σ–p form for η, defined implicitly by (7), we use the chain rule

\frac{\partial z}{\partial η} = (\frac{\partial z}{\partial p}) (\frac{\partial p}{\partial η}),

(60)

then note that hydrostatic balance within the HPEs yields

\frac{\partial z}{\partial p} = - \frac{1}{ρ g},

(61)

so that (59) becomes

\frac{\partial}{\partial t} {(\frac{g_{0}}{g} \frac{\partial p}{\partial η})}_{η} + \nabla_{η} \cdot (v \frac{g_{0}}{g} \frac{\partial p}{\partial η}) + \frac{\partial}{\partial η} (\frac{g_{0}}{g} \dot{η} \frac{\partial p}{\partial η}) = 0,

(62)

after scaling (59) with the constant surface value of g (g₀).

One can immediately see from (62) that one can only proceed to the pure pressure-tendency form of the continuity equation as used in the HPEs by setting g₀/g = 1, consistent with the constant-g approximation routinely adopted within HPE-based models.

This provides a simple proof of the assertion of Deng et al. (2008) that introducing variable g(z) into HPE-based models introduces an inconsistency in mass-wind balance, recognizing that this continuity equation is used subsequently to infer pressure tendencies and associated vertical motion. Put another way, (62) disproves the assertion of Akmaev (2011) that the output of models derived from the HPEs is formally independent of any assumed form for g. In terms of the shallow-atmosphere approximation, White and Wood (2012), extending the earlier work of White et al. (2005), argued that constant g is required in the shallow-atmosphere approximation in order to avoid a spurious nonzero net divergence of the gravitational acceleration vector. Indeed, White and Wood (2012) show that meridional variations in g due to departures from a spherical Earth are in some ways easier to accommodate within the HPEs than vertical variations.

Thus, our approach to using exact height-varying g in a ground-to-thermosphere NAVGEM is similar to the one we settled upon for c_p in section 3g: we use exact profiles on model η surfaces,

g (η) = g_{0} {[1 - \frac{Φ (η)}{g_{0} a}]}^{2},

(63)

in those areas of the code that are diagnostic. These include parameterized physics routines, as well as postprocessing routines that regrid model output onto constant geometric height surfaces, as we have done for many years (e.g., Eckermann et al. 2009, 2018) using

z = \frac{Φ}{g_{0}} {(1 - \frac{Φ}{g_{0} a})}^{- 1} .

(64)

However, we do not follow Akmaev (2011) in using height-dependent gravitational acceleration in the prognostic dynamics. We exclude, for example, his recommendation to convert terrain height h to surface geopotential using the height-dependent form of g as

Φ_{s} = g_{0} \frac{h}{1 + h / a},

(65)

retaining instead the constant-g form (13) for consistency with constant-g HPE approximations.

Our long-term strategy is to transition from NAVGEM to a new forecast model incorporating nonhydrostatic deep-atmosphere forms of the discretized equations that allow variable g to be implemented directly into the core (e.g., Wood et al. 2014).

5. Tests in NAVGEM using idealized globally balanced states

We augmented the dynamical core in NAVGEM with options to run using either MVPT, standard HVPT, or corrected HVPT, then extended the vertical range to ∼200-km altitude (p_top = 5 × 10⁻⁷ hPa) in a 160 level (L160) formulation and to ∼400-km altitude (p_top = 5 × 10⁻¹⁰ hPa) as an L184 configuration.

A range of idealized dynamical core experiments was performed to test the accuracy and robustness of these new vertically extended configurations using the new HVPT-based dynamical core. We present two examples that prescribe an idealized globally balanced state as atmospheric initial conditions. The model is tested for its ability to maintain that balanced state over a 5-day forecast integration.

a. Variable-R thermosphere-like balanced states on nonrotating planet

1) Analytical solutions

We did not find any thermosphere-like variable-R idealized test cases in the literature, so we derived one, as follows. We start with a compatibility equation for a nonrotating planet

\frac{\partial χ}{\partial r} \tan φ + \frac{1}{r} \frac{\partial χ}{\partial φ} = 0,

(66)

derived from the diagnostic relations (11) and (12) of White and Staniforth (2008) and generalized to variable R, such that

χ (φ, r) = \frac{U^{2} (φ, z)}{R (z) T (z)},

(67)

where r = z + a and U is zonal wind. A family of solutions satisfying (66) takes the form

χ (φ, r) = s_{n} (r) \cos^{n} φ,

(68)

where s_n(r) specifies vertical shear and n > 0. Inserting (68) into (66) yields solutions of the form s_n(r) = Arⁿ, where A is constant. Setting

A^{1 / 2} = U_{eq} / \sqrt{R_{0} T_{0}}

, where U_eq is a constant surface equatorial zonal wind, T₀ = T(0) and R₀ = R(0), yields

U (φ, z) = U_{eq} \sqrt{\frac{R (z) T (z)}{R_{0} T_{0}}} {(1 + \frac{z}{a})}^{n / 2} \cos^{n / 2} φ .

(69)

Considering the n = 2 case, the analytical balanced pressure in the absence of terrain is

p (φ, z) = p_{0} \exp [\frac{U_{eq}^{2}}{R_{0} T_{0}} F_{2} (z) \cos^{2} φ - \frac{U_{eq}^{2}}{2 R_{0} T_{0}} \sin^{2} φ - \frac{1}{R_{0} T_{0}} F_{1} (z)],

(70)

where F₂(z) = z/a + z²/2a² and

F_{1} (z) = \int_{0}^{z} \frac{R_{0} T_{0}}{R (z^{'}) T (z^{'})} g (z^{'}) d z^{'}

(71)

is a hydrostatic integral that reduces to geopotential Φ when R(z) = R₀ and T(z) = T₀. For an isothermal, constant composition atmosphere, the zonal velocity has a linear vertical shear that accounts for corotating layers in a deep atmosphere. Simplifying for use with the shallow-atmosphere constant-g NAVGEM HPEs (z/a → 0) yields F₂(z) = 0, whereupon (70) and (71) yield

p (φ, z) = p_{S} (φ) \exp [- g_{0} \int_{0}^{z} \frac{1}{R (z^{'}) T (z^{'})} d z^{'}],

(72a)

p_{S} (φ) = p_{0} \exp (\frac{U_{eq}^{2}}{2 R_{0} T_{0}} \sin^{2} φ) .

(72b)

From (69), the corresponding balanced zonal wind for this shallow-atmosphere n = 2 case is

U (φ, z) = U_{eq} \cos φ \sqrt{\frac{R (z) T (z)}{R_{0} T_{0}}} .

(72c)

This balanced solution provides an excellent test of the ability of HVPT dynamics to maintain a globally balanced thermospheric wind field in which both thermospheric temperature T(z) and specific gas constant R(z) can vary in the vertical, subject to the constraints that (i) the integral in (72a) must exist, and (ii) p → 0 as z → ∞. Although κ does not appear in these solutions, it can be shown that the compatibility relations (66) and (67) and their ensuing balanced solutions hold for atmospheres with any sufficiently smooth vertical variation in κ.

2) Implementation and tests in NAVGEM

Because these solutions are formulated in geometric height rather than pressure, we implement this solution as a NAVGEM initial condition as follows. We use (72b) to initialize p_S, then use gridpoint pressures p(η) on the left hand side of (72a) to solve (72a) iteratively for a corresponding initial z(η), and hence an initial geopotential Φ = g₀z, using Newton’s method with known initial profiles R(z) and T(z). These z(η) values in turn initialize the model temperature T(z), specific gas constants R(z), and zonal winds U(φ, z) from (72c).

To suppress initialization noise through errors in these iterative numerical procedures, we choose analytical functional forms for R(z) and T(z) that provide an analytical solution to the height integral in (72a). We choose piecewise fits for R(z) and T(z), shown in Fig. 3, such that 1/[R(z)T(z)] is quadratic in z above 100 km, and thus analytically integrable. Note in Fig. 3 that the virtual thermospheric temperature $T_{υ_{t}} > 1000 K$ near the L160 model top, providing a stern test of the dynamical core in exposing NAVGEM to virtual temperatures several times larger than heretofore encountered in lower atmospheric NWP.

Fig. 3.

Citation: Monthly Weather Review 151, 8; 10.1175/MWR-D-22-0320.1

To test the ability of the standard HVPT core in NAVGEM to maintain this variable-R balanced state, we set $κ = {\tilde{κ}}_{d} = 2 / 7$ . Figure 4a shows the initial thermospheric virtual temperature $T_{υ_{t}} = [R (z) / {\tilde{R}}_{d}] T (z)$ on the NAVGEM T119L160 grid. Figure 4d shows the initial zonal wind U(ϕ, p) that balances this initial state via (72c). Corresponding plots beneath show zonal-mean errors in NAVGEM virtual temperature and zonal wind after 48 and 120 h of the T119L160 forecast run using standard HVPT with a time step Δt = 120 s. The errors are very small everywhere, with relative errors < 10⁻³. Largest errors accumulate at the upper boundary associated with small amounts of dynamical noise near the model lid.

Fig. 4.

Citation: Monthly Weather Review 151, 8; 10.1175/MWR-D-22-0320.1

The same experiment was repeated using the deeper T119L184 configuration using different vertical profiles for R and T to prevent values from becoming excessively large. The results (not shown) also maintained the analytical balanced wind and temperature states out to 120 h, with largest errors confined to the uppermost few model layers.

As a corresponding test of the ability of the corrected HVPT core in NAVGEM to maintain this state, we allowed κ to vary vertically as $κ (z) = R (z) / {\tilde{c}}_{p_{d}}$ , which influences the forecast via the κ-dependence of the corrected Exner function (48). We solve the locally constant prognostic $θ_{υ_{c}}$ equation given by Eq. (52) since this globally balanced zonal flow should support no vertical motion, so that Dκ/Dt = 0. Figure 5 shows 120-h temperature and zonal-wind forecast errors using the corrected HVPT $θ_{υ_{c}}$ with ς = 1 (top row) and ς = 10⁻⁹ (bottom row). The errors are again small everywhere except in the uppermost few layers. These upper-level errors are noticeably smaller in the ς = 10⁻⁹ forecasts, indicating that reducing the progression toward very large thermospheric $θ_{υ_{c}}$ values in this way tends to reduce overall numerical errors at the upper boundary, in addition to reducing those errors due to omission of Dκ/Dt terms in the prognostic $θ_{υ_{c}}$ equation.

Fig. 5.

Citation: Monthly Weather Review 151, 8; 10.1175/MWR-D-22-0320.1

b. Globally balanced state on rotating planet with thermospheric heating

This test is designed to establish a balanced tropospheric state, then to apply intense thermosphere-like heating above it as the forecast proceeds, to assess the ability of the core to maintain both states at the lower and upper boundaries. As opposed to the previous case, planetary rotation is included.

We initialize the model with the globally balanced initial state of Ullrich et al. (2014), subsequently simplified for a shallow-atmosphere model following their section 4, using a lapse rate of 10⁻⁴ K km⁻¹. To accommodate a deep domain extending into the thermosphere, we modify the initial temperature field at upper levels to asymptote to an isothermal state, which does not affect the balance if this transition occurs at upper levels where the balanced wind response is zero. The blended temperature relation is

T_{blend} (φ, p) = T_{U2014} (φ, p) [1 - β (p)] + T_{iso} β (p),

(73a)

β (p) = \frac{1}{2} {1 - \tanh [2 (\log_{10} p + 3)]},

(73b)

where p is pressure in hPa, T_U2014 is the balanced analytical temperature solution of Ullrich et al. (2014) [their Eq. (20)], and T_iso is the isothermal upper-level temperature. The initial temperature field is given by (73) with T_iso = 300 K and is plotted in Fig. 6a. Since the Ullrich solutions are formulated in height coordinates, implementation of the initial state in NAVGEM involves a hydrostatic remapping from height to pressure coordinates using an iterative numerical solve with a predefined error convergence threshold [see appendix C of Ullrich et al. (2014)].

Fig. 6.

Citation: Monthly Weather Review 151, 8; 10.1175/MWR-D-22-0320.1

As the model integration proceeds, we apply thermospheric heating of the Newtonian relaxational form:

\frac{\partial T}{\partial t} \equiv \frac{Q}{ρ c_{p}} = - α (T - T_{ref}),

(74)

where α is the Newtonian thermal relaxation rate at which the model temperature T is forced toward a reference temperature profile T_ref given by (73) with T_iso = 700 K, as shown in Fig. 6b.

Thus, this idealized experiment establishes a balanced Ullrich et al. (2014) state in the lower atmosphere from the surface to 10⁻² hPa, while applying rapid thermosphere-like heating from 10⁻⁴ to 10⁻⁶ hPa during the forecast. The goal is to see whether the core maintains the balanced tropospheric state while simultaneously maintaining a motionless balanced thermosphere undergoing rapid heating of the isothermal profile. This heating experiment was implemented within NAVGEM and integrated out to five days (120 h) at T119L160 using α = 0.168 56 h⁻¹ (∼6-h e-folding time). The forecast thermosphere is heated to within 1% of the limiting 700-K isothermal state after ∼27.3 h.

The 0-h balanced initial wind fields are plotted in the top row of Fig. 7. Despite imposition of intense thermospheric heating as soon as the forecast commences, these initially balanced winds are maintained throughout the forecast up to high altitudes, where zonal-mean deviations from these 0-h initial conditions after 120 h are <1 m s⁻¹ at all heights and latitudes, and ≪1 m s⁻¹ at heights below the uppermost few model layers.

Fig. 7.

Citation: Monthly Weather Review 151, 8; 10.1175/MWR-D-22-0320.1

These experiments verify that the model maintains a balanced circulation containing a representative idealized troposphere capped by an intensely heated thermosphere. It should be noted that the elastic upper boundary condition of our pressure-based dynamical core allows the thermosphere to expand naturally (via diagnostic increases in $z -$ on a constant pressure surface) as thermospheric heating is applied, permitting a more realistic upper-level response to this intense column heating relative to height-coordinate models using rigid lids [see, e.g., the discussions in Kelly et al. (2023) and Klemp and Skamarock (2022)].

6. Summary and outlook

We have presented a methodology, based around the classical NWP concept of virtual temperature, that allows us to take a preexisting dynamical core designed solely for lower-atmospheric NWP and, via targeted changes that are minimally invasive lower down, to extend it through the thermosphere to capture salient aspects of the variable composition dynamics of the thermosphere. The augmentation is structured around a hybrid virtual potential temperature (HVPT) that asymptotically approaches moist and thermospheric limits in the lower and upper atmospheres, respectively. We implemented these HVPT-based upgrades within the NAVGEM dynamical core, then tested and validated both the standard and corrected HVPT dynamical-core formulations using forecasts of test cases incorporating idealized balanced initial conditions, hot or rapidly heated thermospheres, and height-varying R and κ in the thermosphere.

We are not aware of any previous attempts to generalize virtual temperature to allow lower atmospheric dynamical cores to extend through the thermosphere. One reason could be that moist virtual temperature is a small modification, typically within a percent of dry air temperatures even within highly moist tropospheric environments. By contrast, thermospheric virtual temperatures can be ∼100% larger than local air temperatures in the upper thermosphere. Our work shows that there is no small-perturbation restriction to the virtualization approach, permitting a generalized HVPT that incorporates large changes in thermospheric composition, relative to the lower atmosphere, in ways that accurately capture impacts on adiabatic dynamics throughout the thermosphere.

We consider a major practical advantage of the proposed HVPT modification to be its light footprint on existing tropospheric numerics. The HVPT modification returns the exact MVPT-based equations used for tropospheric NWP by ensuring that $\hat{R} \to 1$ and $κ \to {\tilde{κ}}_{d}$ globally below some “well mixed” boundary level where composition-induced changes to R are no longer relevant (see Figs. 1 and 2). The HVPT modification also adds negligible computational overhead relative to the original MVPT core. Both features make extension of an MVPT-formulated forecast model into the thermosphere using HVPT augmentation more viable operationally, since they minimize potential impacts on existing configurations used in tropospheric NWP. Additional computational cost comes primarily from the inevitable need in any thermospheric extension to add more model layers and to reduce time steps to accommodate the stiffer dynamics of hot thermospheres. While demonstrated here in a hydrostatic core used for operational NWP, these same practical advantages should translate to nonhydrostatic NWP cores that use MVPT-based equation sets for tropospheric prediction (e.g., Davies et al. 2005; Skamarock et al. 2012). HVPT can also augment the θ-based dynamical cores used to model whole-atmosphere climate (e.g., Liu et al. 2018) and the dynamics of other planetary atmospheres of variable composition (e.g., Mayne et al. 2014; Mendonça et al. 2016; Tremblin et al. 2017).

Consistent with our goal of fast-tracking thermospheric NWP development, these and other promising initial findings led us to implement the HVPT dynamical core in a full physics version of NAVGEM running at both T119L160 and T119L184, to provide a background forecast model to accelerate development of NWP-quality thermospheric physics as well as ground-to-thermosphere ensemble data assimilation capabilities within a mature cycling NWP system. This was motivated by an emerging need to drive physics-based ionospheric models with accurate thermospheric forecasts as inputs (e.g., Zawdie et al. 2020). This work will be described more fully elsewhere, but we can report that our prototype HVPT-based forecast model has remained stable and accurate to date, operating now for well over a year at both T119L160 and T119L184 in a variety of research experiments. These include cycling ensemble-based data assimilation experiments extending over months, as well as free-running full-physics nature runs out to 40 days and longer. Thus, our HVPT-based development described here is already achieving its primary goal of fast-tracking ground-to-thermosphere NWP development, as we await a next-generation nonhydrostatic spectral-element forecast model, slated to replace NAVGEM for NWP and currently under active research and development (e.g., Theurich et al. 2016; Zaron et al. 2022). We are concurrently developing high-altitude capabilities for that new model based on new deep-atmosphere nonhydrostatic equation sets without any inbuilt assumptions about composition to admit variable R, κ, and g directly within the dynamical core.

Our validated virtual temperature concepts for the thermosphere also offer promising scientific research diagnostics for understanding thermospheric dynamics, particularly when analyzing zonal-mean or planetary-scale circulations for which constant-κ errors due to vertical motion are small. We leave this for future work, but note a few possibilities, such as thermal-wind and gradient-wind relations based on thermospheric virtual temperature (28) (see, e.g., White and Staniforth 2008), and a thermospheric potential vorticity based on thermospheric virtual potential temperature (see, e.g., Schubert et al. 2001; Schubert 2004; White et al. 2005).

Acknowledgments.

This work was supported by the Defense Sciences Office of the Defense Advanced Research and Projects Agency (DARPA DSO) through their Space Environment Exploitation (SEE) program.

Data availability statement.

IDL code containing all data needed to create the fitted R, c_p,and κ profiles as described in the appendix and to reproduce and plot the profiles in Fig. 1 is available at https://map.nrl.navy.mil/map/pub/nrl/papers/hvpt/. NAVGEM idealized model output data are available from the authors upon request. MSIS data and code are available at https://map.nrl.navy.mil/map/pub/nrl/NRLMSIS/.

APPENDIX

Representative R and κ Profiles for Use in NAVGEM

Since the standard MVPT-based dynamical core adopts constant R and c_p, we sought simple robust initial parameterizations of vertical variations of R and c_p that captured the basic geophysical dependences needed for more realistic forecasts of ground-to-thermosphere dynamics using a vertically extended NAVGEM with a HVPT-based dynamical core. These fits comprise a simple intermediate step prior to implementing more detailed code upgrades to incorporate prognostic thermospheric composition, as required to compute R and c_p values directly.

a. Specific gas constant R

We computed R profiles using temperature and composition from the Mass Spectrometer Incoherent Scatter (MSIS) empirical model (Picone et al. 2002) in a global-mean configuration with diurnal variations included. Examples for a solar F_10.7 = 100 solar flux units (sfu) and geomagnetic A_p = 4 on day 80 (April) at the equator and Greenwich meridian are shown in Fig. A1a, comprising six profiles at equi-spaced universal times spanning a diurnal cycle. As opposed to corresponding profiles plotted versus geometric height (not shown), these pressure-level profiles do not vary appreciably with local time, indicating that diurnal variations result mostly from atmospheric thermal expansion on the hot dayside and contraction on the cold nightside.

Fig. A1.

Citation: Monthly Weather Review 151, 8; 10.1175/MWR-D-22-0320.1

Thus we fitted the diurnal-mean

\hat{R} = R / {\tilde{R}}_{d}

profile with different analytical functions, settling on a sigmoid function for the vertical derivative

d \hat{R} / d Z

, where Z is pressure height (56), since R profiles in Fig. A1a are characterized by a lower boundary condition of dR/dZ = 0 and by an upper boundary condition of constant dR/dZ. Integrating a sigmoid function for

d \hat{R} / d x

, where x = log(1000/p) = Z/H_ref, yields

{\hat{R}}_{fit} (x) = 1 + B (x) a_{0} a_{2} \log {1 + \exp [\frac{(x - a_{1})}{a_{2}}]},

(A1)

where, initially,

B (x) = 1

. The resulting least squares fit

R = {\hat{R}}_{fit} {\tilde{R}}_{d}

is shown as the gold curve in Fig. A1a. Closer inspection of these fits revealed that they did not asymptote sufficiently rapidly to the well-mixed limit of

\hat{R} = 1 (R = {\tilde{R}}_{d})

at levels below 0.01 hPa due to the long asymptotic tails of the sigmoid function. Thus, we apply an additional term,

B (x) = \frac{1}{2} {1 + \tanh [\frac{(x - b_{1})}{b_{2}}]},

(A2)

in (A1) with b₁ = 12.0 and b₂ = 1.8. The final form of this fitted

{\hat{R}}_{fit} (x)

profile is shown in Fig. A1b for three different values of F_10.7.

b. Specific heats c_p and c_υ via κ⁻¹ fits

We utilize the separable form (20) to fit κ⁻¹, which then gives c_p and c_υ by scaling the fitted R profile above. In choosing an appropriate analytical function to fit κ⁻¹, we note that the deep atmosphere is characterized by the diatomic limit $κ_{2}^{- 1} = 7 / 2$ at the surface and the monatomic limit $κ_{1}^{- 1} = 5 / 2$ at the exobase; see Fig. A1c.

Given these properties, we performed fits to a generalized logistic function (aka Richards’s curve):

κ^{- 1} (x) = κ_{2}^{- 1} + B (x) (κ_{1}^{- 1} - κ_{2}^{- 1}) {1 + k_{2} \exp [- k_{1} (x - k_{0})]}^{- k_{3}},

(A3)

where k₀, …, k₃ are the fitted coefficients. As for R we apply the additional function

B (x)

to ensure a faster transition to the diatomic lower-boundary limit

κ^{- 1} = κ_{2}^{- 1}

. Figure A1d shows examples of these fits.

c. Solar cycle variability

These profiles and their associated fitting coefficients are sensitive to solar activity. Solid curves in Fig. A2 show how these coefficients vary with F_10.7. Thicker transparent curves show third-order polynomial fits, which capture the salient F_10.7 variations very well for the $\tilde{R}$ coefficients and acceptably for the κ⁻¹ coefficients. Thus, we have embedded this way of capturing the F_10.7-dependence of the fitted profiles into NAVGEM.

Fig. A2.

Citation: Monthly Weather Review 151, 8; 10.1175/MWR-D-22-0320.1

Table A1 lists the third-order polynomial coefficients fitting the F_10.7 dependence of the least squares fitted coefficients a_j and k_j in Fig. A2. We also fitted κ profiles to the simpler function:

κ (x) = κ_{2} + 0.5 (κ_{1} - κ_{2}) {1 + \tanh [{\tilde{k}}_{1} (x - {\tilde{k}}_{0})]},

(A4)

involving only two fitted parameters

{\tilde{k}}_{0}

and

{\tilde{k}}_{1}

. Cubic polynomial fits to their F_10.7 dependences are also given in Table A1.

Table A1.

Coefficients p_j corresponding to cubic polynomial fits of the form $p_{0} + p_{1} \hat{f} + p_{2} {\hat{f}}^{2} + p_{3} {\hat{f}}^{3}$ , where $\hat{f}$ is F_10.7 (in sfu), to the coefficients a_j, k_j, and ${\tilde{k}}_{j}$ used to fit analytical profiles of R, κ⁻¹, and κ, respectively (see Fig. A2).

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Abstract

Abstract

1. Introduction

2. NAVGEM governing equations

a. Equations of state and hydrostatic balance

b. Pressure gradient force

3. Variable composition and new virtual potential temperature forms

a. Variable gas constants

b. Moist virtual potential temperature (MVPT)

c. Thermospheric virtual potential temperature

d. Hybrid virtual potential temperature (HVPT)

e. Mass specific heats using HVPT θυh

f. Variable κ: A corrected HVPT θυc

1) Specific enthalpy

2) Corrected HVPT

3) Corrected potential temperature and potential enthalpy

4) Standard HVPT errors due to constant κ=κ˜d

g. Diabatic tendency Q

4. Variable gravitational acceleration

5. Tests in NAVGEM using idealized globally balanced states

a. Variable-R thermosphere-like balanced states on nonrotating planet

1) Analytical solutions

2) Implementation and tests in NAVGEM

b. Globally balanced state on rotating planet with thermospheric heating

6. Summary and outlook

Acknowledgments.

Data availability statement.

APPENDIX

Representative R and κ Profiles for Use in NAVGEM

a. Specific gas constant R

b. Specific heats cp and cυ via κ−1 fits

c. Solar cycle variability

REFERENCES

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e. Mass specific heats using HVPT $θ_{υ_{h}}$

f. Variable κ: A corrected HVPT $θ_{υ_{c}}$

4) Standard HVPT errors due to constant $κ = {\tilde{κ}}_{d}$

b. Specific heats c_p and c_υ via κ⁻¹ fits