## 1. Introduction

In meteorology, the laws governing mass, energy, momentum, angular momentum, and entropy, should be sacrosanct. In supercell simulations, invented forces should be tolerated only to the extent that they do not fundamentally alter the results. The latest supercell simulations with surface drag have such high resolution that they reproduce strong to violent tornadoes. These simulations seek to understand tornadogenesis. Use of a fictitious force that generates spurious horizontal vorticity and energy (Roberts et al. 2016, 2020; hereinafter R16 and R20, respectively) obscures conclusions about the physical causes of tornadogenesis.

Difficulties also arise when laws are applied only to a base state and not to deviations from it. For example, mass excesses and deficits arise in supercell models when the continuity equation is satisfied only in the base state. This nonconservation of mass may affect the maximum wind speeds and pressure deficits of tornadoes that form within simulated supercells.

A supercell model solves the governing equations subject to suitable initial and boundary conditions. From a mathematical perspective, the chosen initial conditions are arbitrary. Ideally the initial conditions should not be too far from an equilibrium state as minor changes in initial conditions can result sometimes in very different outcomes. Physically, they should be reasonably realizable. Otherwise, large “shocks to the system” affect the subsequent evolution unrealistically. A model can be initialized with a single environmental sounding as long as the environment is allowed to evolve.

Some modelers, however, desire a specified, unchanging, horizontally uniform environment in which to grow storms. A static environment specified by a single sounding is possible only if the environmental initial state is a steady-state solution of the governing equations. The possibility also exists that a steady solution is unstable. Because of the Taylor–Proudman theorem (Tritton 1988, p. 218), a stipulated static environment is not generally a steady solution. Modelers circumvent this problem by adding a fictitious horizontal force to Newton’s second law of motion. They are then solving the wrong equations. The additional two degrees of freedom associated with the unnatural force allow the environment to be fixed. Unfortunately the required external force, which is a surrogate for the horizontal pressure-gradient force (HPGF) that would exist in a horizontally varying environment, is permanent, pervasive, and a function of height *z* only. Fixing the environment, thus, changes the dynamics inside the storm.

When the invented force is small relative to, say, the HPGF in a mesocyclone, we might expect that its presence is benign. For instance, Wilhelmson and Chen (1982) allowed the Coriolis force to act only on deviations from the environmental wind. This is equivalent to using an invented force that cancels the Coriolis force in the environment.

However, above the PBL, the invented force has to replace the large-scale HPGF and balance the Coriolis force in order to maintain a static horizontally uniform environment. Use of the external force instead of the HPGF uncouples the horizontal equation of motion from the other equations governing the environment. In the presence of vertical wind shear and Coriolis forces, a real balanced environment must have a horizontal thermodynamic gradient. Strongly sheared supercell environments are baroclinic, not horizontally uniform. In an unrealistic, static, horizontally uniform environment, there is no baroclinity and the Rossby number (ratio of inertial to Coriolis forces) is zero because the horizontal length scale is infinite. Under these conditions, the Taylor–Proudman theorem dictates that the wind shear must vanish above the PBL if the model is run without a storm. If we allow a horizontal density gradient, then we find below that a steady sheared environment with horizontally uniform winds is possible only if the atmosphere above the PBL is equivalent barotropic, which is still uncharacteristic of supercell environments.

The invented force can cause graver problems when there is friction in the environment. When the lower boundary condition is free slip, the stratification is stable enough, and turbulence is not introduced artificially, zero eddy viscosity is a solution of the model sans storm (Markowski and Bryan 2016, hereinafter MB16). But serious difficulties arise in simulations that implement surface drag and hence activate friction near the ground. To prevent diminishing environmental winds, supercell modelers allow surface drag to affect only the wind deviation (e.g., Wilhelmson and Chen 1982; Adlerman and Droegemeier 2002). This amounts to inventing a force **K**(*z*) that cancels the environmental frictional force **F**_{0} and the environmental Coriolis force **C**_{0} (if present), thus, enabling a static horizontally uniform environment. If the modeler does not impose significant random perturbations to excite turbulence and the stratification is sufficiently stable, the parameterization scheme that simulates the effects of turbulence on subgrid scales is active in the environment only in a very shallow surface layer (MB16). Then the shear stress, which is specified at the ground by the semislip boundary condition, vanishes at a low height above the ground. Consequently **F**_{0}, which is the vertical derivative of shear stress, is large in a shallow surface layer. Since **K** cancels **F**_{0} and **C**_{0}, it too is large next to the ground. This is evident in Fig. 8 in Dawson et al. (2019, hereinafter DRX19) where **K** exceeds 10^{−2} m s^{−2} near the ground. Inside the storm, **K**(*z*) is the same as outside of it but is no longer balanced by the frictional and Coriolis forces. This unbalanced artificial force harms the simulation.

Tornadogenesis depends critically on near-surface processes so the presence of a significant artificial unbalanced force **K**(*z*) near the ground is not conducive to its understanding. Because **K** balances the frictional and Coriolis forces in the environment, it is affected greatly by the model’s parameterization of turbulence. Unfortunately, supercell models do not perform well next to the ground for three reasons. First, because of use of a staggered grid, the horizontal resolved wind is not defined at the ground itself. The lowest level at which it is defined is one-half of a vertical grid spacing above the ground (the “lowest scalar level”). Second, large-eddy simulation (LES) has a known weakness next to the ground where the dynamically important eddies are unresolved (Davidson 2015, p. 402; discussed further in section 3). Third, as demonstrated by MB16, the vertical profile of the frictional force in LES is unrealistic if random perturbations with sufficient amplitude are not introduced into the flow to excite significant turbulence.

As shown herein, an invented force is clearly no panacea because it manufactures false accelerations, false energy, false horizontal vorticity, and false contributions to circulations around inaccurately calculated material circuits. [Circuits that dip below the lowest scalar level are not trusted even without an invented force (Dahl et al. 2012)]. Conclusions about tornadogenesis drawn from recent simulations using a permanent external force therefore seem questionable. These studies conclude that that a tornado may originate from frictional rather than baroclinic vorticity. Obviously part of the vertical vorticity of a rotating updraft ultimately is generated by environmental frictional torques. It stems from vorticity generated by the shear stress at the ground, diffused into the PBL, and ultimately tilted and stretched by the updraft. Friction is also vital in the contraction of a tornado cyclone into a tornado that breaks the thermodynamic speed limit (Fiedler and Rotunno 1986). Thus, frictional interaction of the airflow with the ground is believed to play an important role in tornadogenesis. However, parameterized frictional torques may be generating too much horizontal vorticity in too shallow a surface layer, thus, making tornadogenesis too rapid. As shown by Rotunno et al. (2017), uplifting of excessive near-ground horizontal vorticity results in significant vertical vorticity at very low heights.

## 2. Unnatural initialization of supercell models

For reasons listed by DRX19 (4035–4036) and R20 (p. 1703), modelers often covet an ideal, static, horizontally uniform environment. Consequently, many supercell models use a single sounding for initialization and set up a steady environmental flow without horizontal gradients by using an artificial force. In contrast, the observed wind profile in the PBL has a flow component toward low pressure, so the atmospheric environment varies horizontally. Without imposition of an external force, a horizontally uniform environment with Coriolis force and vertical wind shear is never in equilibrium because of thermal-wind imbalance above the PBL. The environment evolves by developing secondary circulations that tend to restore geostrophic balance (Holton 1992, section 6.4).

For a simulation initialized with a single sounding, establishing thermal-wind balance with a vertical shear of around 10^{−2} s^{−1} requires a linear horizontal temperature gradient of around 3 K (100 km)^{−1}. Replacing the environmental HPGF with an invented force is advantageous because it eliminates the need for this temperature gradient. An actual sounding already incorporates the effect of friction in the Ekman layer. However, the parameterized friction and heat fluxes in the model are imperfect matches to atmospheric turbulence. Without an environmental HPGF, the model’s surface drag and internal frictional forces would rapidly modify the environmental wind profile over time, especially near the ground, as kinetic energy would dissipate rapidly. Another advantage of an external force is that it performs work to negate this energy loss. We shall see, however, that the invented force also has its downsides.

There are three recent techniques to incorporating surface drag into simulations. The first two methods use a spurious horizontal force to constrain the environment to be static and horizontally uniform. The artificial force added to Newton’s second law of motion introduces two extra degrees of freedom, thus, allowing the supplemented model equations to have superfluous solutions. The first approach (R16; Roberts and Xue 2017, hereinafter RX17; Coffer and Parker 2017), called the frictional balancing procedure (FBP) by R20, takes a proximity hodograph representative of the environment and modifies it (Figs. 1 and 2). This procedure is discussed further in section 7.

The second approach (DRX19; Oliveira et al. 2019; R20), which supersedes the first, is called the geotriptic wind balance (GWB) method. It uses a horizontally uniform environment that is hydrostatically balanced and in three-way equilibrium between the Coriolis force, the parameterized turbulent frictional force, and an invented force **K**(*z*) (Figs. 1 and 3). This method is evaluated in section 8.

The third approach utilizes nested models and so does not start from a horizontally uniform state. Since it does not use an invented force, it is beyond the scope of this paper. We should note, however, that the tornadoes in these simulations do not occur early in a storm’s lifetime and that diagnostics find that the vorticity origins of many of the tornadoes are frictional rather than baroclinic (e.g., Schenkman et al. 2014; Mashiko 2016; Tao and Tamura 2020).

Early storm development is influenced by convective initiation as well as by the specified environment. Convection is often initialized in a supercell model by releasing a warm bubble at low levels. A large bubble is used to overcome any convective inhibition and in environments without a cap to obtain an isolated long-lived supercell instead of narrow towering cumuli that cannot withstand the shear (Davies-Jones 2002). The hope is that, in the long run, storm structure becomes independent of the initial conditions, namely the way that the environment was upset to trigger the storm. The bubble is inserted into the initial conditions as a strong temperature perturbation without any initial adjustment whatsoever to the wind field and the vortex lines. In an extreme case, a wide bubble with a large 6-K amplitude in a very unstable environment with a lifted index of −10.5 K and a lot of streamwise vorticity establishes an intense rotating updraft atypically soon. The environment that is intended to remain horizontally uniform and steady is upset by gravity and sound waves radiating outward from the rising thermal (as exemplified by Fig. 4). As the updraft intensifies at low levels, high winds, abnormally strong low-level convergence, and a tornado intensified by the effort of surface drag, all develop at the surface within 30 min of convective initiation.

## 3. Short description of supercell models

*f*plane in the Northern Hemisphere (where the Coriolis parameter

*f*> 0) are

*t*is time,

**v**is the wind vector,

*d*/

*dt*≡

*∂*/

*∂t*+

**v ⋅**∇ is the material derivative,

*ρ*is density,

*p*is pressure,

*θ*is potential temperature,

*g*is the gravitational acceleration,

**F**is the divergence of the turbulent subgrid-scale momentum flux per unit mass (loosely called the frictional force), and

**K**(

*z*) is a spurious horizontal force. The rate of potential temperature change per unit mass is

*δQ*/

*δt*is the rate that heat is added per unit volume,

*D*is the divergence of the turbulent theta-flux vector,

*R*is the gas constant, and

*c*

_{p}is the specific heat at constant pressure. The equation of state is

*γ*≡

*c*

_{p}/

*c*

_{υ},

*c*

_{υ}is the specific heat at constant volume, and

*p*

_{00}is a standard pressure (1000 hPa). The entropy

*S*=

*c*

_{p}ln

*θ*up to an additive constant, and the temperature

*T*=

*θ*(

*p*/

*p*

_{00})

^{κ}, where

*κ*=

*R*/

*c*

_{p}. In terms of eastward, northward, and vertical unit vectors

**i**,

**j**,

**k**, respectively, the position vector is

**x**≡

*x*

**i**+

*y*

**j**+

*z*

**k**≡

*x*

_{1}

**i**+

*x*

_{2}

**j**+

*x*

_{3}

**k**, and the wind vector is

**v**≡

*u*

**i**+

*υ*

**j**+

*w*

**k**≡

*u*

_{1}

**i**+

*u*

_{2}

**j**+

*u*

_{3}

**k**.

**v**,

*ρ*,

*p*, and

*θ*. The prognostic equation for mass in (2) is often replaced by a combination of (2) and (3), namely,

*parameterized*subgrid-scale term

*D*via (3). Since there are no sound waves to filter in a steady environmental flow and we want accurate mass conservation that does not involve parameterized heat fluxes, we generally retain (2).

**v**:

**F**, and

**K**.

**≡ ∇ ×**

*ζ***v**is

*α*≡ 1/

*ρ*is the specific volume and −∇

*α*× ∇

*p*, the curl of the PGF, is the baroclinity vector

**B**(Dutton 1976, p. 374). From (12) and (2) the steady-state vorticity equation for an environment with small Rossby number is

**K**≡ 0, 1) solenoidal generation of horizontal vorticity must cancel the horizontal vorticity produced by tilting of Earth’s vertical vorticity by the shear flow and 2) solenoidal generation of vertical vorticity must cancel the density-advection term in (13). Conclusion 2 holds even with an invented force

**K**(

*z*) since its curl has no vertical component.

*C*(

*t*) yields the circulation equation:

**K**, this is the circulation theorem derived by Dutton (1976, p. 373).

*C*

_{D}= [

*κ*/ln(Δ

*z*/2

*z*

_{0})]

^{2}is the drag coefficient,

*z*

_{0}is the roughness length, and

*κ*≈ 0.4 is von Kármán’s constant (MB16). The free-slip condition is

*K*

_{m}is the eddy viscosity and

*K*

_{h}is the eddy diffusivity for heat. Following Deardorff (1972),

*K*

_{m,}supercell models typically use a 1.5-order closure scheme involving the TKE (Emanuel 1994, 290–297). The eddy viscosity is related to TKE by

*c*

_{k}is a constant (typically 0.1) and Δ is an SGS length scale. Usually Δ = (Δ

*x*Δ

*y*Δ

*z*)

^{1/3}where Δ

*x*, Δ

*y*, and Δ

*z*are the grid spacings in the

*x*,

*y*, and

*z*directions.

*ε*. The constant

*c*

_{ε}is typically 0.1. By definition, TKE cannot be negative so

*K*

_{m}by solving (23) and (22). A trivial solution is

A weakness in the LES approach becomes apparent next to the ground. There the dynamically important eddies, which are very small near a solid boundary, are not resolved by the grid (Davidson 2015, p. 402). Furthermore, the mechanism of vorticity generation at the ground is not represented in LES. Physically, it is the *viscous* shear stress at the surface that generates vorticity, which then diffuses away from the boundary into the fluid (Davidson 2015, p. 51, 132, 403).

## 4. The Taylor–Proudman theorem

A steady, horizontally uniform environment without a heat source and Coriolis force is a solution of the governing equations for any wind profile **v**(*z*) if the lower boundary condition is free slip and the environment is turbulent free. On an *f* plane the situation is very different. The Taylor–Proudman theorem (Tritton 1988) only permits certain wind profiles. Adding an external force **K**(*z*) lifts the constraint by admitting superfluous solutions. We derive the theorem here since severe-storm modelers may not be familiar with it.

The Taylor–Proudman theorem is valid provided that the flow is quasi-steady, the Rossby number and Ekman number (ratio of frictional to Coriolis force) are much less than 1, the baroclinity vector **B** is small relative to the Coriolis parameter times the vertical wind shear (de la Torre Juárez 2009), and **K** ≡ 0. Except close to the ground, all these conditions are met in the steady horizontally uniform environment of a supercell model prior to convective initiation.

_{H}is the horizontal-gradient operator and the second term vanishes by mass conservation in (2) for steady flow. When

**K**(

*z*) ≡ 0,

*ρ*

**v**is independent of height. In a constant-density flow,

*f∂*

**v**/

*∂z*= 0. This is the Taylor–Proudman theorem, which states that in an incompressible or Boussinesq flow with

*f*≠ 0 and

**K**(

*z*) ≡ 0, the vertical shear must vanish for a steady solution to exist. This constraint explains the phenomenon of Taylor columns (Tritton 1988, p. 218). Note that (27) is just a slight generalization of the theorem. The Taylor–Proudman theorem is circumvented if either the environment is unsteady or extra degrees of freedom are manufactured.

## 5. Steady environment with winds that vary only with height

*static*(

*∂*/

*∂t*= 0) environment without a heat source (

*δQ*/

*δt*= 0) and with horizontally uniform winds

**v**

_{0}(

*z*) =

*u*

_{0}(

*z*)

**i**+

*υ*

_{0}(

*z*)

**j**. The solutions are valid in a neighborhood of (

*x*,

*y*) = (0, 0). We shall find once again that solutions exist only for specific conditions. Since environmental friction depends on the wind only, it is of the form

**F**

_{0}(

*z*) ≡

*G*

_{0}(

*z*)

**i**+

*H*

_{0}(

*z*)

**j**. The invented force

**K**(

*z*) ≡

*L*

_{0}(

*z*)

**i**+

*M*

_{0}(

*z*)

**j**by definition. Each thermodynamic field consists of a vertically varying base state, denoted by subscript 0, and a deviation, which is smaller by at least an order of magnitude, with subscript 1. Thus,

**v**=

**v**

_{0}(

*z*) and (28) into the governing equations in (1)–(4) yields

**K**is a surrogate for the HPGF so

**K**≠ 0 is used only when HPGF = 0 and vice versa. The steady-state continuity equation in (31) does not permit density advection when the environmental winds are horizontal uniform. The entropy equation in (32) represents a balance between advection and diffusion of entropy. When

**K**≡ 0, the set (29)–(33) constitutes an overdetermined system of five equations in the three unknowns

*p*

_{1,}

*ρ*

_{1}, and

*θ*

_{1}. This system has solutions only in special circumstances.

*x*,

*y*) = (0, 0), we may assume that the fields are linear in

*x*and

*y*. Thus, we let

*a*(

*z*) → 0 and

*b*(

*z*) → 0 as

*z*→ ∞. Then from the hydrostatic equation in (30) and the equation of state equation in (33):

*D*to

*D*

_{0}(

*z*) because all other terms in (39) are functions of just height.

When *a* = *b* = *D*_{0} = 0 and **K** ≠ 0, the system (37)–(39) always has a solution. This is the basis of the GWB method (section 8). We next consider the case in which *p*_{1} is nonzero and **K** ≡ 0. Even though **K** ≡ 0, we retain it for completeness.

*ρ*

_{1}= 0 to examine. Then from (35)

*da*/

*dz*=

*db*/

*dz*= 0, which satisfies (38). Setting

*da*/

*dz*=

*db*/

*dz*= 0 in

*d*/

*dz*of (37) produces

*d*(

*ρ*

_{0}

**v**

_{0})/

*dz*must vanish in the free atmosphere when

*f*≠ 0 and

**K**≡ 0. This is simply the generalized Taylor–Proudman theorem of section 4.

*ρ*

_{1}> 0. Inserting (37) into (38) yields

The quantity *υ*_{0}*du*_{0}/*dz* − *u*_{0}*dυ*_{0}/*dz* is the ground-relative helicity (GRH) density (Davies-Jones et al. 1990). Hence, only wind profiles without GRH density are allowed in the free atmosphere when *f* ≠ 0 and **K** ≡ 0. In contrast a tornado environment has a great deal of GRH above the PBL (Fig. 5) as demonstrated in the next section. Zero GRH density (shear vector parallel to ground-relative wind) occurs in an equivalent barotropic atmosphere, and is aptly referred to as inactive baroclinicity (Haltiner and Martin 1957, p. 409). Above the PBL, the isobars, isopycnics, and isentropes all must lie along the wind direction, which must be invariant with height because of **v**_{0} ⋅ (29) and (31)–(33). Otherwise, the solenoidal term generates vertical vorticity [see (12)] and hence horizontal gradients of the horizontal wind, which violates the assumption that **v** = **v**_{0}(*z*).

**v**

_{0},

**v**

_{0}⋅

**F**

_{0}< 0 [see the diagram in Haltiner and Martin (1957), p. 237]. Therefore positive work done by either the HPGF or the external force is the process that offsets frictional loss of kinetic energy in the environment. This means that for steady flow in the PBL the wind must have a component toward low pressure if there is an HPGF (Holton 1992, p. 133). From (42) and (31)–(33),

**v**

_{0}⋅ ∇

_{H}

*ρ*

_{1}, is no longer constrained to be zero. By substituting for the advection terms via (42) and (41) (without the leading “0 = ”), we find that the entropy equation in (32) after use of (33) provides the following condition for a solution:

**K**≡ 0 exist in the free atmosphere only when the GRH density is zero there.

## 6. Tornadic-storm environments

*f*plane at small Rossby number are

**= −**

*ω***i**∂

*υ*/∂

*z*+

**j**∂

*u*/∂

*z*is the relative-vorticity vector. Through suitable choice of the invented force

**K**(

*z*), we can obtain a horizontally uniform environment that has the same GRH density as an observed hodograph. Via conversion to pressure coordinates, the hydrostatic equation and the equation of state (

*p*=

*ρRT*), we obtain the following relationship:

_{p}is the operator

**i**(

*∂*/

*∂x*)

_{p}+

**j**(

*∂*/

*∂y*)

_{p}. Then (46) becomes

*h*

_{1}and

*h*

_{2}nominally define the effective base and top of the storm’s inflow layer (typically

*h*

_{1}= 0 and

*h*

_{2}= 3 or 1 km). The GRH is minus 2 times the area swept out from

*z*=

*h*

_{1}to

*h*

_{2}by the ground-relative wind vector

**v**(

*z*) (Fig. 5; Davies-Jones et al. 1990). Note that the area swept out by the geostrophic wind is proportional to geostrophic warm-air advection (Haltiner and Martin 1957, p. 209). GRH is additive. In other words, the GRH for the 0–

*h*

_{2}layer is equal to the sum of the 0–

*h*

_{1}and

*h*

_{1}–

*h*

_{2}GRHs. From (48) and (49), the GRH is

^{−1}contributes approximately 100 m

^{2}s

^{−2}of GRH per 1 km of inflow depth. We find the GRH associated with an Ekman layer by neglecting thermal advection and

**K**, assuming constant eddy viscosity so that

**F**=

*K*

_{m}(

**i**

*d*

^{2}

*u*/

*dz*

^{2}+

**j**

*d*

^{2}

*υ*/

*dz*

^{2}), and plugging in the standard Ekman-layer solution (Holton 1992, 129–130). The result is

**v**

_{g}is the geostrophic wind. For |

**v**

_{g}| = 14 m s

^{−1}, the GRH of the Ekman layer is 100 m

^{2}s

^{−2}.

**v**−

**c**, where

**c**is the storm motion vector. It is a measure of how favorable the environmental wind profile is for tornado development (Davies-Jones et al. 1990; Droegemeier et al. 1993). SRH is defined by

*z*=

*h*

_{1}to

*h*

_{2}by the storm-relative wind vector (Fig. 5; Davies-Jones et al. 1990). The integrand in (52) is the SRH density. It is an important quantity because it is linked theoretically to overall updraft rotation. In an updraft with a quasi-circular cross section, the covariance of vertical velocity and vertical vorticity is proportional to it (Davies-Jones 1984). From (52) and (51), the SRH is

**v**

_{0}(

*h*

_{2}) −

**v**

_{0}(

*h*

_{1}) is the

*h*

_{1}–

*h*

_{2}shear vector. The last term in (53) is the part of SRH due to storm motion to the right of this shear vector and the integral is the GRH. Like GRH, SRH is additive.

Figure 5 shows an example of a proximity hodograph for a violent tornado that occurred in the warm sector of a cyclone. In this case, the GRH for the 0–1-km layer is 91 m^{2} s^{−2} over the nominal height of the PBL (0–1 km) and is much higher, 439 m^{2} s^{−2}, for the 0–3-km layer. The 0–1- and 0–3-km SRH values, 330 and 570 m^{2} s^{−2}, respectively, are even higher because of storm motion to the right of the corresponding shear vectors. In this case as well as in the mean tornado-outbreak proximity hodograph [see Fig. 11 of Maddox (1976)], large GRH and veering wind direction above the PBL indicate that an atmosphere favorable for tornadoes is far from equivalent barotropic.

There is not a large 0–1-km GRH in Fig. 5 because there is not much near-ground warm-air advection in a warm sector. For storms just ahead of warm fronts, there is large environmental GRH near the surface from strong warm advection (Maddox et al. 1980). The geostrophic warm-air advection contribution to GRH is also associated with synoptic-scale upward motion and airmass destabilization (Holton 1992). A significant invented force is necessary to compensate for lack of thermal advection if the environment is forced to be horizontally uniform.

Equation (50) for GRH has only thermal advection and friction terms because we have assumed small Rossby number. Many tornado outbreaks are associated with small-scale ageostrophic circulations that change rapidly on a time scale of a few hours. Our assumption of small Rossby number is invalid at these scales. Such ageostrophic circulations may form in association with shortwave troughs, jet streaks, low-level jets, boundaries, mesolows, dryline bulges, upslope winds, and diurnal wind oscillations (Maddox 1993). They can set the stage for tornado development by swiftly increasing low-level wind veering and helicity. Thus, significant amounts of GRH in a hodograph may arise from ageostrophic winds. Modeling these situations really requires a mesoscale model.

## 7. The frictional balancing procedure

The frictional balancing procedure assumes falsely that the winds in the actual hodograph are geostrophic at all heights. It also neglects the horizontal temperature gradient associated with thermal-wind balance. The negative of the Coriolis force acting on the environmental geostrophic wind is the external force in this method (Fig. 2). Adding this force is equivalent to letting the Coriolis force act only on deviations from the environmental geostrophic wind. The model is then run without a storm until it reaches a steady state. The ageostrophic part of the Coriolis force is now in balance with the frictional force (Fig. 2). The ageostrophic wind so determined is added to the “geostrophic hodograph” to form an “input hodograph,” which is steady because there is a three-way force balance in its PBL between the external, the Coriolis, and the parameterized frictional forces.

*p*

_{1}=

*a*=

*b*= 0 the system in (37)–(39) has the following solution:

**v**

_{g}are the winds in the geostrophic hodograph and

**v**

_{ag}are the additional winds needed to form the input hodograph

**v**

_{0}(

*z*) from the geostrophic one. In FBP, the external force

**K**cancels the geostrophic part of the Coriolis force. The FBP hodograph modification reduces the ground-relative surface wind and creates a significant ageostrophic wind. These changes enable the Coriolis force acting on the ageostrophic wind to balance the frictional force at the ground as required by (55). We can deduce the frictional force from the difference in the two hodographs in Fig. 1 of R20.

**K**from (56) using GWB (section 8), and (unnecessarily) determine

**v**

_{g}from the ambiguous (55). As long as the same hodograph is input to the model, as is the case with the R16 and RX17 FBP simulations, and the R20 GWB simulations, GWB subsumes FBP and renders it redundant.

## 8. The geotriptic wind balance method

The geotriptic wind balance method (GWB) purportedly uses a horizontally uniform environment that is hydrostatically balanced and in three-way equilibrium between the Coriolis force, the parameterized turbulent frictional force, and the HPGF. Unfortunately, such equilibrium is strictly impossible in a *sheared horizontally uniform* environment with a Coriolis force because, above the PBL, the wind shear is in thermal-wind balance with the large-scale *horizontal temperature gradient*, which has been set to zero. In a hydrostatic environment, the hydrostatic equation and the ideal gas law link the pressure and virtual temperature fields to the horizontal momentum equation. To preserve a horizontally uniform steady environment, “the large-scale PGF must somehow be specified and *decoupled* from the actual model pressure field” (DRX19, p. 3936). To accomplish this, an external force again is used that enables the environmental HPGF to vanish. The external force in GWB is the negative of the resultant of the Coriolis force acting on the environmental wind and the environmental friction (Fig. 3). By adding the external force permanently to the horizontal equation of motion, the environment is maintained throughout the simulation.

**K**is called the pseudo–pressure-gradient force (PPGF) because it replaces the HPGF in the three-force balance (Fig. 3). Introducing this force provides two more degrees of freedom

*L*

_{0}and

*M*

_{0}and, thus, allows solutions for all wind profiles. It neglects the horizontal temperature gradient associated with thermal-wind balance (R20, p. 3375). In the environment there is only geostrophic wind balance in the free atmosphere and the three-force balance within the PBL. It is easily verified that (57) and

*p*

_{1}=

*ρ*

_{1}=

*D*

_{0}= 0 satisfy (29)–(32), which govern the steady environment.

To obtain a 3D steady, horizontally uniform environment, DRX19 (p. 3938) and R20 ran the model without initiating a storm and without the external force. Since **K**(*z*) is the force needed to equilibrate the Coriolis and frictional forces, it is then determined as minus the horizontal average of the initial tendency of **v**_{H}. Including **K**(*z*) establishes a steady-state solution for any environmental wind profile. Since (32) implies *D*_{0} = 0 in steady, horizontally uniform flow, the resulting PBL is well mixed. Figure 1 shows the balance of forces in the environment above the PBL. Because **K** counterbalances both the environmental frictional force and the Coriolis force acting on the environmental wind, *f* is in effect zero for flow in the undisturbed steady environment. The Taylor–Proudman theorem (section 4), which only applies to flows with small Rossby and Ekman numbers, is undone because the Rossby number for the environment is now effectively infinite instead of much less than one. Inclusion of the external force has eliminated the nagging Taylor–Proudman theorem (and the thermal wind and Taylor columns for good measure). The replacement of the HPGF by the PPGF **K** has a dramatic effect. It decouples the horizontal equation of motion from the hydrostatic equation, thus, allowing the much coveted, steady, horizontally uniform environment to be attained. There are no longer horizontal gradients or advection of any thermodynamic variable in the environment. The significant GRH density above the PBL that exists in tornado environments (section 6) is kept in the model artificially through the force substitution. From the energy perspective, (42) indicates that in GWB the spurious work rate **v**_{0} ⋅ **K** has to offset the frictional loss of environmental kinetic energy −**v**_{0} ⋅ **F**_{0}. Work performed by the environmental HPGF would supply this energy in real tornado environments.

In GWB the environmental wind profile is immutable by the decree that declares the environmental forces are always in equilibrium. This is true even when resolved turbulence is present because its effects on the wind profile are nullified by then varying the PPGF at every time step to maintain the equilibrium (DRX19, p. 3945). (In this case the made-up force is a function of *t* as well as *z*.)

A solution of (23) is TKE = 0, which signifies that the SGS turbulence scheme is inactive. This can happen in the environment of simulations with the free-slip lower boundary condition. Since there are no eddies (MB16, p. 1848), resolution of important eddies near the ground is not an issue. To obtain a deep turbulent boundary layer in neutral stratification with surface drag, MB16 introduced random temperature perturbations of significant amplitude (0.25 K) to promote vertical mixing. With surface drag and without excitation of significant turbulence, there is a danger in GWB that the environment’s neutral boundary layer is too shallow and that the shear in the lowest few hundred meters is too large (MB16).

**K**(

*z*) and its associated torque

*d*

**K**/

*dz*everywhere. Within the storm, the frictional and Coriolis forces do not counterbalance the permanent and pervasive PPGF. Eliminating

**K**from (1) via (57) exposes the deleterious effect of this procedure. We obtain

**K**also affects the circulation analyses. R20’s simulations with a nonzero drag coefficient produce early tornadoes. The surface drag causes circulation gain around a material circuit by retarding near-surface parcels whose velocity is contributing negatively to the circulation while having little effect on higher parcels in the circuit. To examine the effect of GWB on this result, we substitute (57) into (14) to eliminate

**K**. This yields the following circulation theorem for GWB:

**K**term in (14) has been dismissed as an “artifact.” However, the last term in (59) reveals that the main consequence of

**K**is to effectively change the frictional-force circulation to the line integral of

**F**−

**F**

_{0}(

*z*) instead of just

**F**. Considering only the line integral of

**F**can lead to false conclusions about the role of surface drag in tornadogenesis. Because the bogus force

**K**also alters the path and configuration of the material circuit

*C*(

*t*), results of the circulation analyses seem doubtful.

Hence the spurious circulation [the last term in (14)] cannot be dismissed. Figure 11 of R20 shows the (positive) circulation around a material circuit and the production terms as functions of drag coefficient from time *t* = 23 min, when the circuit is a horizontal low-level circle of radius 1.5 km encircling the developing tornado back to *t* = 13 min. Since **K** is a function just of *z*, the artificial term vanishes when a circuit is flat. This behavior is evident in the R20 Fig. 11. At the end of the backward trajectories at 13 min., the spurious production term is positive in all the experiments and it is the largest in all but one. By reasonable extrapolation of the curves to the initial time *t* = 0, it seems probable that the artificial term produces most of the circulation in the simulations with no to moderate surface drag.

## 9. Conclusions

Without an externally imposed force, a steady environment with horizontally uniform winds is possible only if the winds in the free atmosphere are parallel to the isobars, isotherms, and isopycnics. Because a horizontal temperature gradient is accompanied by vertical shear and entropy is conserved, these conditions only hold for winds that are unidirectional above the friction layer and consequently for environments with little ground-relative helicity. With vertical wind shear and Coriolis force, a steady, horizontal uniform solution is possible for any wind profile *only if* the horizontal equation of motion becomes decoupled from the hydrostatic equation. This is achieved through the use of an artificial force, which enables the Taylor–Proudman theorem to be circumvented.

Maintaining an ideal, steady, horizontally uniform, sheared environment in a supercell model with Coriolis and frictional forces comes at a cost to the dynamics. The most recent method introduces a permanent omnipresent made-up horizontal force **K**(*z*) that accelerates all parcels all the time. It balances the model’s Coriolis and horizontal frictional forces prior to convective initiation. It performs illusory work that in the prestorm environment offsets kinetic energy dissipated by friction. It unnaturally uncouples the environmental horizontal equation of motion from the hydrostatic equation and severs the dynamics from the thermodynamics. Addition of the extraneous force has the same effect as changing the local frictional force from **F** to **F** − **F**_{0}(*z*), where **F**_{0}(*z*) is the frictional force at the same height *z* in the environment, and similarly altering the Coriolis force.

Moreover ∇ × **K**(*z*) fabricates excessive horizontal vorticity very close to the ground so there is also a substantial spurious circulation. **K** also changes the parcel trajectories, which further affects the material vorticity and circulation analyses.

In recent simulations with surface drag and powerful initiation, tornadoes develop from frictionally generated vorticity atypically early in the lifetime of their parent supercells and far from precipitation. Although vortices such as waterspouts and landspouts occur early in the life cycles of their parent clouds, they form in environments that are not horizontally homogeneous. Occasionally tornadoes form in young supercell storms, but little is known about their mesoscale environments and their causes. Revealingly, low-precipitation supercells produce relatively few tornadoes (Davies-Jones et al. 1976; Bluestein and Parks 1983). If supercell tornadoes form just from low-level cloud-scale convergence and rotation plus surface drag, they would occur much more frequently and generally much earlier in the life cycle of the parent supercell.

Rapid tornadogenesis in simulations may be due to a combination of the following factors:

excessively strong convective initiation in a very unstable atmosphere, resulting in too strong a low-level updraft too early in a supercell’s lifetime,

use of the semislip lower boundary condition,

absence of perturbations to excite significant turbulence, resulting in too strong a frictional force and too much shear in too shallow a layer next to the ground (MB16), and

use of an omnipresent, pervasive external force to maintain a steady, horizontally uniform environment; this force introduces extra degrees of freedom that allow the Taylor–Proudman theorem to be circumvented but has the unintended consequence of altering the dynamics inside the storm.

A possible mechanism for rapid tornadogenesis in a simulation is as follows. As a result of factor 4, the external force is fixed by the three-force balance in the environment. It is concentrated near the ground by factors 2 and 3. Inside the storm, friction and the Coriolis force no longer balance it so it generates large amounts of horizontal vorticity in a shallow surface-based layer. Tilting and stretching of this vorticity (if it happens to be streamwise) by the updraft in factor 1 will produce significant vertical vorticity at low heights (as demonstrated by Rotunno et al. 2017).

The advices for supercell modelers are threefold. First, do not initiate convection violently and expect early storm development to be realistic. Second, introduce random perturbations of sufficient strength to obtain a deep turbulent PBL. Third, do not resort to an artificial force that decouples the horizontal momentum and hydrostatic equations and generates spurious energy and vorticity.

## Acknowledgments

Valuable comments by Drs. Paul Markowski and Louis Wicker and the three anonymous reviewers prompted significant improvements in the paper.

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