## 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*, 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

_{υ}*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*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.

_{υ}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

*R*,

*c*,

_{p}*c*, and

_{υ}*g*. Quantifying any two of the first three variables gives the third via Meyer’s relation:

*η*(defined below). Here

*t*is time,

*λ*is longitude,

*φ*is latitude,

*a*is Earth’s radius,

**∇**=

*a*

^{−1}[cos

^{−1}

*φ*(∂/∂

*λ*), ∂/∂

*φ*],

*f*is the Coriolis parameter,

**v**= (

*u*,

*υ*) is horizontal wind vector, (

*F*,

_{u}*F*) are momentum forcing terms,

_{υ}*Q*is diabatic heating rate (expressed as time rate of change of energy density),

*p*is pressure,

*p*is surface pressure,

_{S}*ρ*is density, Φ is geopotential,

*p*

_{0}is a constant reference-level pressure (typically 1000 hPa), and

*T*are explained in section 3 when variable composition is considered.

*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)

*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.

### a. Equations of state and hydrostatic balance

*α*is specific volume. The geopotential

*g*

_{0}= 9.8066 m s

^{−2}is the constant value of

*g*at the surface,

*z*is geometric height, and

*dp*= −

*ρgdz*yields

*h*is terrain height.

### b. Pressure gradient force

*η*-surface gradients using [e.g., (1-44) of Haltiner and Williams (1980)]

*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

*i*= 1, …,

*I*

_{tot}mass constituents with individual mass mixing ratios

*r*and specific gas constants

_{i}*c*is evaluated using ideal-gas relations as a mass-mixing-ratio-weighted sum of the individual species terms:

_{p}*β*is the total number of degrees of freedom of gas species

_{i}*i*,

*m*is its mass, and

_{i}*k*is the Boltzmann constant. Defining a species volume mixing ratio

_{B}*q*and noting that

_{i}*R*, viz.

*β*= 5). Since ∼99% of the atmosphere from 0 to ∼100 km comprises the diatomic molecules N

_{i}_{2}and O

_{2}, then the sum in (20a) is approximately constant:

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*in Fig. 1c are lower, due to the accompanying decrease in

_{p}*κ*

^{−1}with height in Fig. 1b.

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)

*ρ*and

_{d}*ρ*, respectively, yields

_{w}*r*is the mass mixing ratio of water vapor. Since

_{w}*R*can differ from the constant dry-air value

_{m}*p*(and hence exact pressure gradient forces) via the equation of state:

The key point here is that by virtualizing our prognostic potential temperature variable *R*, despite retaining a constant “dry” value of

*R*within the geopotential pressure integral, yielding exact moisture-modified geopotentials, and hence exact pressure gradient forces −

_{m}**∇**

*Φ via (17) in the momentum equations.*

_{p}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

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

*R*is a thermospheric specific gas constant calculated using (18) assuming no moisture contribution (

_{t}*s*= 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.

_{w}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*that can vary in all three spatial dimensions and in time, just like

_{t}*s*in (24).

_{w}### d. Hybrid virtual potential temperature (HVPT)

*R*s are normalized by the constant lower atmospheric value

*R*s are constants, whereas untilded

*R*s vary in space and time, viz.

*R*away from

_{t}*T*in the stratosphere and mesosphere, and the thermospheric virtual temperature

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

_{w}### e. Mass specific heats using HVPT ${\theta}_{{\upsilon}_{h}}$

*c*and

_{υ}*c*variabilities are included implicitly through

_{p}*κ*, which in turn implies variable

*c*and

_{p}*c*since

_{υ}*R*now varies. For each

*R*profile in Fig. 1a, dotted curves in Fig. 1c show the corresponding

*c*profiles in Fig. 1c since, as shown in Fig. 1b,

_{p}*κ*

^{−1}decreases with height.

### f. Variable κ: A corrected HVPT ${\theta}_{{\upsilon}_{c}}$

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

#### 1) Specific enthalpy

*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

*T*and virtual temperature

*c*, gives the pure temperature form

_{p}*κ*,

*R*, and

*c*are all exact. This equation can now be rearranged into two equivalent forms: a mixed

_{p}*c*is locally constant (

_{p}*Dc*/

_{p}*Dt*= 0), whereupon

*κ*is locally constant (

*Dκ*/

*Dt*= 0). Thus, the supposed shortcomings of using

*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*, (41) and (43) still insert a virtualized temperature variable into these equations to play the same pivotal role in converting the constant

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

*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

*ς*is set to unity for now.

The major change to the virtual potential temperature definition in (47) is the *D* log*κ*/*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*is

*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

A new *Dκ*/*Dt* term appears in (49), needed to remove an identical term contained within the material derivative of *ς* = 1, the log*ς* term in square parentheses in (49) vanishes, leaving a −log(*p*/*p*_{0}) 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).

*Dκ*/

*Dt*term in (49) can be minimized to make a locally constant approximation of

*Dκ*/

*Dt*≈ 0 more tenable. Using the standard

*p*

_{0}= 1000 hPa, the −log(

*p*/

*p*

_{0}) correction term in (49) becomes large and positive in the thermosphere, since

*p*/

*p*

_{0}≪ 1. The

*ς*factor in (48) allows us to choose a modified

*ς*= 10

^{−9}yields

^{−6}hPa, a nominal pressure level where

*κ*is undergoing greatest vertical variations in Fig. 1b. At

*p*=

^{−6}hPa, the

*Dκ*/

*Dt*term in (49) vanishes and the solution is exact, while at pressures a decade above and below

*p*

_{0}= 1000 hPa (

*ς*= 1). While these changes make

*Dκ*/

*Dt*term in (49) vanishes anyway below ∼10

^{−4}hPa (cf. Fig. 1b). While

*ς*= 10

^{−9}, note from (49) that these estimated error reductions apply to

*Dκ*/

*Dt*, implying smaller relative errors in

*Dκ*/

*Dt*= 0 as a more tenable approximation in (49) gives the compact equation:

*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

*p*= 0, in which case (17) could continue to be used since these expressions remain exact on terrain-following tropospheric layers, since

_{S}#### 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* = *T*Π^{−1} as the prognostic variable, then derive *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

*c*variability and so both require a corrected HVPT for more accurate adiabatic temperature solutions. Incorporating corrected HVPT into models that predict

_{p}*θ*requires migrating to a corrected potential temperature

*θ*-specific Exner correction

*θ*equation of either the general form (49) or approximated form (52), such that

_{c}*θ*and Π

_{c}*now replace*

_{c}*c*dependence in (54b) results from the conversion of

_{p}*T*to

*θ*now provides an indirect solution to the potential enthalpy equation (cf. Juang 2011). Prognostic

_{c}*θ*is then converted diagnostically into standard HVPT

_{c}*θ*-based dynamical core to provide an indirect potential enthalpy solution that improves the prediction of variable-

*c*thermospheres.

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

What other typical errors do we anticipate when using a globally constant

*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*and

_{t}*κ*, is given by

*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

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

^{−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

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*from assuming constant

_{s}*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 *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*is the exact profile computed from (19). As discussed earlier and as can be shown using (37), when using

_{p}*c*in (42) is replaced everywhere by the uncorrected HVPT form of

_{p}*c*given as

_{p}*c*for the tendency

_{p}*Q*/

*ρc*to ensure a proper diabatic heat budget in the thermosphere. Once completed, we convert temperature back to HVPT prior to the next dynamics step.

_{p}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 cos*φ* Coriolis terms, yield greater errors by comparison (White and Bromley 1995).

*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.

*η*, mass continuity takes the general form [Eq. (3.21) of Kasahara 1974]:

**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

*g*(

*g*

_{0}).

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 *g*_{0}/*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.

*g*in a ground-to-thermosphere NAVGEM is similar to the one we settled upon for

*c*in section 3g: we use exact profiles on model

_{p}*η*surfaces,

*h*to surface geopotential using the height-dependent form of

*g*as

*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

*R*idealized test cases in the literature, so we derived one, as follows. We start with a compatibility equation for a nonrotating planet

*R*, such that

*r*=

*z*+

*a*and

*U*is zonal wind. A family of solutions satisfying (66) takes the form

*s*(

_{n}*r*) specifies vertical shear and

*n*> 0. Inserting (68) into (66) yields solutions of the form

*s*(

_{n}*r*) =

*Ar*, where

^{n}*A*is constant. Setting

*U*

_{eq}is a constant surface equatorial zonal wind,

*T*

_{0}=

*T*(0) and

*R*

_{0}=

*R*(0), yields

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

*F*

_{2}(z) =

*z*/

*a*+

*z*

^{2}/2

*a*

^{2}and

*R*(

*z*) =

*R*

_{0}and

*T*(

*z*) =

*T*

_{0}. 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

*F*

_{2}(

*z*) = 0, whereupon (70) and (71) yield

*n*= 2 case is

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 =

*g*

_{0}

*z*, 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

To test the ability of the standard HVPT core in NAVGEM to maintain this variable-*R* balanced state, we set *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 Δ*t* = 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.

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*-dependence of the corrected Exner function (48). We solve the locally constant prognostic *Dκ*/*Dt* = 0. Figure 5 shows 120-h temperature and zonal-wind forecast errors using the corrected HVPT *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 *Dκ*/*Dt* terms in the prognostic

### 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.

^{−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

*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)].

*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.

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

## 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 *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*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

_{p}*R*and

*c*values directly.

_{p}#### 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.

*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

*x*= log(1000/

*p*) =

*Z*/

*H*

_{ref}, yields

*b*

_{1}= 12.0 and

*b*

_{2}= 1.8. The final form of this fitted

*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*by scaling the fitted

_{υ}*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

*k*

_{0}, …,

*k*

_{3}are the fitted coefficients. As for

*R*we apply the additional function

#### 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 *κ*^{−1} coefficients. Thus, we have embedded this way of capturing the *F*_{10.7}-dependence of the fitted profiles into NAVGEM.

*F*

_{10.7}dependence of the least squares fitted coefficients

*a*and

_{j}*k*in Fig. A2. We also fitted

_{j}*κ*profiles to the simpler function:

*F*

_{10.7}dependences are also given in Table A1.

Coefficients *p _{j}* corresponding to cubic polynomial fits of the form

*F*

_{10.7}(in sfu), to the coefficients

*a*,

_{j}*k*, and

_{j}*R*,

*κ*

^{−1}, and

*κ*, respectively (see Fig. A2).

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