Editorial Type:
Article
Article Type:
Research Article

Balanced Convective Circulations in a Stratified Atmosphere. Part I: A Framework for Assessing Radiation, the Coriolis Force, and Drag

David H. Marsico aDepartment of Mathematics, University of California, Davis, Davis, California

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Joseph A. Biello aDepartment of Mathematics, University of California, Davis, Davis, California

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Matthew R. Igel bDepartment of Land, Air and Water Resources, University of California, Davis, Davis, California

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Online Publication:
12 Dec 2023
Print Publication:
01 Dec 2023
DOI:
https://doi.org/10.1175/JAS-D-22-0254.1
Page(s):
2915–2924
Free access

Abstract

The so-called traditional approximation, wherein the component of the Coriolis force proportional to the cosine of latitude is ignored, is frequently made in order to simplify the equations of atmospheric circulation. For velocity fields whose vertical component is comparable to their horizontal component (such as convective circulations), and in the tropics where the sine of latitude vanishes, the traditional approximation is not justified. We introduce a framework for studying the effect of diabatic heating on circulations in the presence of both traditional and nontraditional terms in the Coriolis force. The framework is intended to describe steady convective circulations on an f plane in the presence of radiation and momentum damping. We derive a single elliptic equation for the horizontal velocity potential, which is a generalization of the weak temperature gradient (WTG) approximation. The elliptic operator depends on latitude, radiative damping, and momentum damping coefficients. We show how all other dynamical fields can be diagnosed from this velocity potential; the horizontal velocity induced by the Coriolis force has a particularly simple expression in terms of the velocity potential. Limiting examples occur at the equator, where only the nontraditional terms are present, at the poles, where only the traditional terms appear, and in the absence of radiative damping where the WTG approximation is recovered. We discuss how the framework will be used to construct dynamical, nonlinear convective models, in order to diagnose their consequent upscale momentum and temperature fluxes.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: David H. Marsico, dhmarsico@ucdavis.edu

Abstract

The so-called traditional approximation, wherein the component of the Coriolis force proportional to the cosine of latitude is ignored, is frequently made in order to simplify the equations of atmospheric circulation. For velocity fields whose vertical component is comparable to their horizontal component (such as convective circulations), and in the tropics where the sine of latitude vanishes, the traditional approximation is not justified. We introduce a framework for studying the effect of diabatic heating on circulations in the presence of both traditional and nontraditional terms in the Coriolis force. The framework is intended to describe steady convective circulations on an f plane in the presence of radiation and momentum damping. We derive a single elliptic equation for the horizontal velocity potential, which is a generalization of the weak temperature gradient (WTG) approximation. The elliptic operator depends on latitude, radiative damping, and momentum damping coefficients. We show how all other dynamical fields can be diagnosed from this velocity potential; the horizontal velocity induced by the Coriolis force has a particularly simple expression in terms of the velocity potential. Limiting examples occur at the equator, where only the nontraditional terms are present, at the poles, where only the traditional terms appear, and in the absence of radiative damping where the WTG approximation is recovered. We discuss how the framework will be used to construct dynamical, nonlinear convective models, in order to diagnose their consequent upscale momentum and temperature fluxes.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: David H. Marsico, dhmarsico@ucdavis.edu
Keywords: Subgrid-scale processes; Convective-scale processes; Dynamics; Differential equations

1. Introduction

The full Coriolis force contains terms proportional to the sine and cosine of latitude. The former are referred to as the traditional Coriolis terms and couple the zonal and meridional momentum equations. The latter, referred to as the nontraditional Coriolis terms (NCTs), couple the zonal and vertical momentum equations. Scaling arguments have often been used to justify the neglect of the NCTs. For instance, in midlatitude, synoptic-scale meteorology, it can be shown that the nontraditional Coriolis term in the zonal momentum equation is relatively small, and in the vertical momentum equation, it is negligible compared to vertical accelerations, gravity, and the vertical pressure gradient. Under these circumstances, the “traditional approximation” is made, whereby the NCTs are neglected, but the traditional Coriolis terms (TCTs) are retained. However, near the equator, the cosine and sine of latitude approach unity and zero, respectively, and it becomes more difficult to justify the outright neglect of the nontraditional terms for circulations which are not in hydrostatic balance.

The effect of the nontraditional Coriolis terms has been studied in different contexts. They have been considered in convection (Igel and Biello 2020), tropical waves (Ong and Roundy 2020; Ong and Yang 2022), convective momentum transport (LeMone 1983), oceanic dynamics (Marshall and Schott 1999), and idealized studies of the planetary boundary layer (Dubos et al. 2008). The work of Igel and Biello (2020) shows how the NCT and the pressure field induced by convective circulations create a purely horizontal force which acts on the circulation. In the framework described below, this horizontal force will manifest as a secondary horizontal circulation added to the primary convective circulation. The nontraditional Coriolis terms have also shown to be important in shallow water approximations (Stewart and Dellar 2013, 2012, 2010). In addition, a set of equations that retain the nontraditional Coriolis terms and possess conservation principles for mass, energy, and potential vorticity were derived in Tort and Dubos (2014). However, it is largely case that the influence of the NCTs on atmospheric flows remains incompletely understood and poorly appreciated. Studies of the nontraditional terms tend to conclude that, when considered diligently, the NCTs should not be ignored in low-latitude meteorological situations with the potential for or the occurrence of sustained vertical motion.

Our original intention for this work was to study the NCTs only in a broad way. We wanted to introduce a mathematical framework for understanding tropical dynamics under the influence of the NCTs that would be applicable from the synoptic scales to the mesoscales and would not necessarily invoke wave dynamics, the latter having been the focus of most previous work on the NCTs. To do so, we introduced a scaling of the incompressible Euler equations on an equatorial beta plane that would allow us to study the NCTs’ effect on the corresponding steady-state equations. However, we realized that our analysis could easily be extended to the Euler equations at an arbitrary latitude, and the case where only the nontraditional terms are present could be obtained by evaluating the theory at zero latitude.

To yield a general, albeit linear, framework, we consider the impacts of radiation and dissipation of momentum on the dynamics. The latter allows the possibility of steady-state solutions. Consideration of the former is motivated by mesoscale studies of tropical systems which tend to emphasize the important role of radiation, especially in horizontal gradients of radiative heating (Wing et al. 2017), and by its fundamental role in the energy balance of the tropical atmosphere (Manabe and Strickler 1964). As a consequence of our choice of time and length scales, and in the absence of radiation, there is a simplification of our equations that yields one of the fundamental features of the weak temperature gradient (WTG) approximation: the direct diagnosis of vertical velocity from the heating. The WTG approximation has been applied on mesoscales and synoptic scales in the tropics to understand, among other things, tropical cyclone formation (Raymond et al. 2007; Adames et al. 2021), the Madden–Julian oscillation (Chikira 2014), and the Walker cell (Bretherton and Sobel 2002). At first glance, it may be counterintuitive that convection can be described by a diagnostic equation for the vertical velocity since it is understood to be achieved on meso- and synoptic scales in the tropics. However, balance of the form of WTG requires that the waves travel across the region of interest more quickly than the circulation transports the fluid. In this framework, the gravity wave travel time across an isolated convective element is much faster than a convective turnover time, which are the time scales under consideration. This time-scale separation means that gravity waves quickly restratify the potential temperature (or buoyancy) in the vicinity of the convection so that the time derivative of the buoyancy equation can be neglected in favor of its balanced state (a radiation modified version of WTG). A WTG balance on convective scales was first developed by Klein and collaborators and was summarized nicely by Klein (2010). More recently, a diagnostic equation for the vertical velocity in deep convection was also derived by Hittmeir and Klein (2018) using the method of asymptotic scale analysis.

The derivation of our framework will begin with a nondimensionalization and scale analysis but will set aside a systematic asymptotic analysis for the future. We split our work into two parts. Here, in Part I, we derive sets of diagnostic equations for velocity, pressure, and buoyancy perturbation. We consider three distinct cases to elucidate the effect of the Coriolis force on convective flows: when the full Coriolis force, only the nontraditional terms, or only the traditional terms are retained. The last two cases occur at the equator and pole, respectively. Since the equatorial, nontraditional Coriolis case is of the most interest to us, it is presented fully in Part II (Marsico et al. 2023, manuscript submitted).

This paper is organized as follows. In section 2, we discuss the velocity and time scales for which the incompressible Euler equations yield solutions corresponding to equilibrated circulations on atmospheric convective length scales, as would be used for subgrid convective parameterizations in large-scale computations. Since this is a preliminary framework, we focus on flow strengths that can be described by linear theory because they are weak enough. The effects of turbulent dissipation on subgrid scales are often approximated by drag damping, or enhanced, turbulent diffusivity. In our model, we will use linear dissipation on convective scales to account for the enhanced diffusivity associated with subgrid turbulence. We also focus on time scales where the zonal and meridional components of the full Coriolis force balance the pressure gradients and damping, while the vertical component balances the vertical pressure gradient, damping, and buoyancy.

To solve the resulting steady linear equations, it is necessary to introduce damping, and we consider two forms: first, constant drag damping in the momentum equations and Newtonian cooling in the buoyancy equations; second, diffusive damping in the momentum equations and Newtonian cooling in the buoyancy equation. In section 3, we use the Helmholtz decomposition to separate the velocity field into two components. The poloidal component of the velocity field is horizontally convergent and directly responds to the heating; we thus describe it as the primary circulation (Zhang and Schubert 1997). A purely horizontal velocity field is generated from the poloidal circulation, the Coriolis force, and the momentum damping; we describe it as the secondary circulation.

There are two significant physical predictions of our framework regarding the effect of NCT and radiation. The first is expressed by Eq. (18), which arises as a balance between the “net Coriolis force” (Igel and Biello 2020) and momentum damping. It provides a simple relationship between the vertical derivative of the streamfunction of the secondary circulation and the derivative of the potential function of the primary (poloidal) circulation along the axis of rotation of Earth. The second is expressed in Eq. (19), where the potential of the primary, poloidal circulation is related to the latent heating through an elliptic operator. In the absence of radiation, this expression reduces to the weak temperature gradient approximation; that is to say, the vertical velocity is proportional to the latent heating. Radiation allows the effect of latent heating to be felt away from its source, thereby providing a mechanism for descent or ascent away from the center of convection. In section 4, we contrast solutions to these equations at the equator (purely NCT) versus the poles (purely TCT). In section 5, our results are summarized.

2. Length and time scales of the primitive equations appropriate to convective circulations

Our framework describes steady, convective circulations under the influence of buoyancy, NCT, TCT, and damping. In this and our companion manuscript, the framework will be linear. Our reasoning is that nonlinearity will primarily create turbulent dissipation (modeled as a linear damping) and can be mostly accounted for by eddy diffusivity. Future work will extend these results to circulations where advective nonlinearities cannot be neglected, yet the weak temperature gradient will be maintained. It is the versatility of the WTG simplification that allows for simple solutions in both linear and nonlinear steady circulations. Furthermore, in the linear regime, the various properties of the circulation and buoyancy response to diabatic heating can be straightforwardly associated with their sources and sinks, making this framework a natural starting point for a dynamical convective parameterization.

In the following paragraphs, we nondimensionalize the equations of motion and describe the relevant spatial, temporal, velocity, and buoyancy scales. Although we will ultimately work with a linear and dimensional model, the discussion of nondimensionalization is important to ensure our framework remains consistent with flows we seek to describe. Furthermore, we envision this framework as the first step toward a multiscale analysis of the nonlinear effects of convection on meso- and synoptic-scale circulations in keeping with Klein (2010), Hittmeir and Klein (2018), and Hirt et al. (2023). A careful multiscale analysis must begin with a clear nondimensionalization of the equations of motion in order to identify the relevant small parameters used in the asymptotic method. Therefore, with an eye to future applications, we proceed with the scale analysis.

We begin with the incompressible, stratified, damped Euler equations on an f plane at a latitude λ,
∂u∂t+u⋅∇u−2Ωυsin⁡(λ)+2Ωcos⁡(λ)w=−∂ϕ∂x−d1u,
∂υ∂t+u⋅∇υ+2Ωusin⁡(λ)=−∂ϕ∂y−d1υ,
∂w∂t+u⋅∇w−2Ωucos⁡(λ)=−∂ϕ∂z+b−d1w,
DbDt+N2w=S,
∇⋅u=0,
where b = gθ/θ0 is the buoyancy perturbation, θ is the potential temperature perturbation, θ0 is a reference potential temperature, d1 is the damping coefficient due to the sub-cloud-scale turbulent dissipation (or damping operator if, e.g., a drag parameterization is used), N2=(g/θ0)(dθ˜/dz) is the squared buoyancy frequency of the unperturbed atmosphere, θ˜(z) is the background potential temperature stratification, and ϕ = p/ρ0 + gz is the Montgomery potential for a constant density fluid ρ0. The buoyancy source is related to the diabatic heating through S = (g/θ0)Sθ. Since we consider an idealized theoretical framework, we use the incompressible Eq. (1e), instead of the anelastic continuity equation.
To nondimensionalize the equations, we introduce the length, time, velocity, buoyancy, pressure, and latent heating scales (L, T, U, b0, ϕ0, S0), as follows: (x, y, z) = L(x′, y′, z′), t = Tt′, (u, υ, w) = U(u′, υ′, w′), b = b0b′, ϕ = ϕ0ϕ′, and S = S0S′. Since the scaling is isotropic in the vertical and horizontal directions, the resulting vertical momentum equation will not express hydrostatic balance. Instead we allow for the possibility that all of the linear forces participate in the dominant balance at lowest order. Rewriting Eqs. (1a)–(1e) in terms of the nondimensional variables (and dropping primes for readability), we find
∂u∂t+UTLu⋅∇u−2ΩTsin⁡(λ)υ+2ΩTcos⁡(λ)w=−ϕ0TLU∂ϕ∂x−d1Tu,
∂υ∂t+UTLu⋅∇υ+2ΩTsin⁡(λ)u=−ϕ0TLU∂ϕ∂y−d1Tυ,
∂w∂t+UTLu⋅∇w−2ΩTcos⁡(λ)u=ϕ0TLU(−∂ϕ∂z+b0Lϕ0b)−d1Tw,
b0N2UT(∂b∂t+UTLu⋅∇b)+w=S0N2US,
∇⋅u=0.
As with all asymptotically inspired methods, one attains a simplified model by seeking a dominant balance between different terms in the primitive equations. However, the vertical and horizontal length scales under consideration are fixed by the troposphere height. Choosing L = 7 km allows for deep convective circulations (order 2L) as well as developing convection (order L/2).

The Coriolis force participates in the dominant balance when 2ΩT ≥ 1, which means that we consider time scales of T = (2Ω)−1 ≈ 2 h or larger. Notwithstanding that on a 2-h time scale, the time derivatives in the momentum equation may not necessarily be negligible, the balanced circulations we consider herein can be thought of as either the equilibration of a convective circulation under Coriolis and damping, or a quasi-stationary, slowly evolving circulation pattern due to latent heating.

The relative strength of the nonlinear terms to the linear terms is measured by the Rossby number
UTL=U2ΩL≡Ro.
A linear regime is applicable if the Rossby number of the flow is less than one. So Ro < 1 implies the velocity U is less than the scale 2ΩL ≈ 1 m s−1. From the perspective of small-scale turbulent motions in atmospheric convection, this is indeed a small velocity. However, we expect that this velocity scale is appropriate to the large-scale envelope of convection and that the smaller-scale, faster motions contribute to the subcloud enhanced turbulent diffusion.
Buoyancy-driven circulations of low Mach number (the ratio of the characteristic speed to the speed of sound) result in incompressible (or anelastic) velocity fields to a high degree of approximation, and this is maintained by the pressure gradient. Therefore, we expect that the buoyancy and pressure perturbation are the same order of magnitude, b0 = ϕ0/L. Indeed, this is often observed in simulated active convection (Jeevanjee and Romps 2016; Peters 2016). We also expect that the pressure gradient and buoyancy be on the same order of magnitude as the Coriolis force, thereby ϕ0 = (UL)/T = Ro(L2)/T2, which yields the buoyancy scale
b0=RoLT2.
Using UT = RoL, we can estimate the coefficient multiplying the temperature transport term on the left side of Eq. (2d) to be
b0N2UT=RoLT21N2Ro L=(NT)−2≡ϵ2,
where the last equality is the definition of ϵ. Since the Brunt–Väisälä frequency in the troposphere is approximately N = 0.02 s−1 and by using a Coriolis time scale T ≈ 7200 s, we find
ϵ≈1144,
so that the temperature advection term on the left-hand side of Eq. (2d) is extremely small compared to the vertical transport of the background stratification [the w term on the left-hand side of Eq. (2d)]. Effectively, this means that gravity waves are extremely fast compared with advection. Therefore, the weak temperature gradient approximation, where the vertical velocity balances the diabatic heating in a diagnostic equation, is an excellent approximation even on convective scales. We also expect that the momentum damping will balance the Coriolis force, so that the damping rate d1−1 is of order T ≈ 2 h.

These scale arguments establish the time, length, and diabatic heating scales for which the linear, steady approximation provides an excellent description of the circulation. Convective circulations do not necessarily satisfy these constraints throughout their development, but the linear steady theory can still provide insights into the induced circulation, even if nonlinear advection would tend to slowly evolve such a circulation.

Radiative damping

We are ultimately interested in the effect that radiative cooling has on steady-state circulations and can model its effect by introducing a Newtonian cooling term of the form −d2b to the right-hand side of Eq. (1d). This term would then be nondimensionalized as −(d2b0)/(N2U)b on the right-hand side of Eq. (2d). Inclusion of this radiative term in no way changes any of the previous scaling arguments. Now, if the diabatic heating source and radiative sink on the right-hand side of the temperature equation are to be in balance with the vertical velocity, then S0 ≈ d2b0 ≈ UN2 = Ro × 1 m s−1 × (0.02 s−1)2 = 1.44 m s−2 h−1. At the small buoyancy perturbations considered here, this balance requires a somewhat large Newtonian cooling parameter d2.

3. Linear convective WTG with full Coriolis force

In this section, we derive the framework of the linear convective WTG with the full Coriolis force. As we discussed above, we consider the linear, steady versions of Eqs. (1a)–(1e) with a heating source and linear cooling in the temperature equation and damping in the momentum equations. For the momentum equations, we will discuss both linear drag and enhanced turbulent diffusion.

To elucidate the physics of the problem, as well as simplify the mathematics, we will begin by using the Helmholtz decomposition to separate the horizontally convergent flow which directly responds to diabatic heating from the horizontally nonconvergent flow which arises as a balance between the Coriolis force and the momentum damping. The theory will consist of an elliptic (Poisson-like) equation for the horizontal velocity potential with a source term given by the diabatic heating. We will show how the other variables, the horizontal streamfunction, pressure, buoyancy, and the three components of the velocity, can all be diagnosed from this velocity potential. The linear convective WTG equations with Coriolis force are
−2Ωsin⁡(λ)υ+2Ωcos⁡(λ)w=−∂ϕ∂x−d1u,
2Ωsin⁡(λ)u=−∂ϕ∂y−d1υ,
−2Ωcos⁡(λ)u=−∂ϕ∂z+b−d1w,
N2w=S−d2b,
∂u∂x+∂υ∂y+∂w∂z=0,
where d1 is the momentum damping coefficient and d2 is the radiative damping coefficient. We first consider the case when d1 and d2 are due to Newtonian drag and radiative damping and then show how the theory can be easily extended to account for turbulent diffusion.
To describe the analytic solution of these equations, we follow the Helmholtz decomposition (Helmholtz 1867; Lebovitz 1989), introducing the streamfunction ψ and velocity potential Φ, and write the velocity field as
u=(−Φx−ψy)i+(−Φy+ψx)j+wk.
The horizontally irrotational component, described by Φ, can converge in the horizontal direction [it was described as horizontally confluent in Igel and Biello (2020)] and constitutes a poloidal vector field which is directly tied to the vertical velocity through a kinematic expression. The horizontally rotational component is described by a streamfunction ψ and therefore has no convergence in the horizontal plane. Its relationship to the velocity potential is a consequence of physics, as we will describe below. Setting the divergence of Eq. (4) to zero yields the well-known Poisson equation for the velocity potential in terms of the vertical velocity:
∇h2Φ=wz,
where ∇h2 is the Laplacian operator in the horizontal (x, y) direction alone. Taking the vertical component of the curl of the velocity field yields the (also) well-known expression of the streamfunction in terms of the vertical component of vorticity
∇h2ψ=vx−uy.
We now derive an equation for Φ in terms of S by eliminating the pressure and buoyancy from the momentum equations. We will then use Eqs. (4)–(6) to write this equation in terms of Φ. Differentiating Eq. (3a) with respect to z and Eq. (3c) with respect to x and eliminating ϕ yield
d1uz−2Ωsin⁡(λ)υz+2Ωcos⁡(λ)wz=−2Ωcos⁡(λ)ux−bx+d1wx.
Differentiating Eq. (3b) with respect to z and Eq. (3c) with respect to y and eliminating ϕ yield
2Ωsin⁡(λ)uz+d1υz=−2Ωcos⁡(λ)uy−by+d1wy.
Now, we use Eq. (3d) to eliminate b from Eq. (7)
d1uz−2Ωsin⁡(λ)υz+2Ωcos⁡(λ)wz=−2Ωcos⁡(λ)ux+N2wx−Sxd2+d1wx
and from Eq. (8)
2Ωsin⁡(λ)uz+d1υz=−2Ωcos⁡(λ)uy+N2wy−Syd2+d1wy.
Notice d2 appears in the denominator in both Eqs. (9) and (10), and this term would be singular if d2 were zero. In this limit, the WTG approximation is recovered for w and therefore Φ, i.e., N2w = S. Upon differentiating Eq. (9) with respect to x and Eq. (10) with respect to y, adding the results, taking the z derivative, and making some rearrangements, we obtain
d1(uxz+υyz)+2Ω[sin⁡(λ)(uyz−υxz)+cos⁡(λ)(wxz+∇h2u)]=(N2+d1d2)∇h2w−∇h2Sd2.
Using the incompressibility constraint, Eq. (11) simplifies to
−d1d2wzz−2Ω d2[sin⁡(λ)(υxz−uyz)+cos⁡(λ)(υxy−uyy)]=(N2+d1d2)∇h2w−∇h2S.
One can recognize the vertical component of vorticity, υx − uy, in both Coriolis terms on the left-hand side of Eq. (12), which we will eliminate in favor of the Laplacian of the streamfunction. The vorticity is operated on by the derivative
∂∂n≡cos⁡(λ)∂∂y+sin⁡(λ)∂∂z.
This is the directional derivative parallel to the direction of the north polar axis (thus our choice of “n”) as viewed from the tangent plane at latitude λ. A way to visualize this derivative is that, at a latitude λ, the derivative is taken in a direction that points toward the North Star. Taking the derivative of Eq. (12) with respect to z, we can use Eq. (5) to replace w in favor of Φ, and upon rearranging the expression, we find
(N2+d1d2)∇h4Φ+d1d2∇h2Φzz=∇h2Sz−2Ω d2∇h2ψnz.
Next, we invert one instance of the horizontal Laplacian throughout Eq. (14). The resulting expression would have an arbitrary harmonic function, which is the kernel of the Laplacian, added to the right-hand side. However, all harmonic functions either grow at infinity (corresponding to solutions growing away from the source) or are singular at a point in the domain (corresponding to solutions that blow up at a point). Therefore, we can set the harmonic function to zero, and we arrive at one expression which relates the horizontal streamfunction, the horizontal convergence (potential Φ), and the diabatic heating
(N2+d1d2)∇h2Φ+d1d2Φzz=Sz−2Ω d2 ψnz.
We have chosen to work with the potential for the horizontal convergence Φ in order to attain an expression which does not contain any horizontal derivatives of S. In the companion paper, we consider diabatic heating profiles with horizontal discontinuities, such as would be expected during cloud formation, and wish to avoid second derivatives of discontinuous functions. By setting d2 = 0, we recover the WTG approximation from Eq. (15).
To construct a single elliptic PDE for Φ, we need another expression relating the streamfunction to the potential. Note that the derivatives we have used to arrive at Eq. (15) construct the horizontal components of the vorticity equation. Subtracting the y derivative of Eq. (3a) from the x derivative of Eq. (3b) eliminates the horizontal pressure gradient and describes the vertical component of the vorticity equation, which is not directly affected by buoyancy,
2Ω[sin⁡(λ)(ux+υy)−cos⁡(λ)wy]=−d1(υx−uy).
Taking the z derivative of Eq. (16) and replacing the components of the velocity with the streamfunction and potential,
−2Ω[sin⁡(λ)∇h2Φz+cos⁡(λ)∇h2Φy]=−d1∇h2ψz.
Again, using the expression for the directional derivative along the north polar axis, inverting an instance of the horizontal Laplacian on each term, and swapping the sides of the equality yield the extremely simple relationship relating the streamfunction to the velocity potential:
d1∂ψ∂z=2Ω∂Φ∂n.
Equation (18) is elegant, deceptively simple, and merits some elucidation. Although the right-hand side is measured in units of acceleration, it arose from the vertical torque due to the Coriolis force acting on a poloidal velocity field described by Φ (Igel and Biello 2020). From the Helmholtz decomposition, the poloidal component of a velocity field is uniquely determined from its vertical component, yielding the convergence in the horizontal plane which compensates for the vertical circulation; that is to say, it is the solution of Eq. (5) substituted into Eq. (4). This is a significant relationship between the convective, primary circulation described by Φ and the horizontal, secondary circulation described by the streamfunction ψ. Its derivation was motivated by the computation in Igel and Biello (2020) of the divergence free portion of the Coriolis force induced by a convective velocity field. When this divergence free component of the Coriolis force is balanced by momentum drag (or dissipation), Eq. (18) results.

The left-hand side of Eq. (18) arises from the damping of the vertical component of the vorticity. That vertical component of vorticity is, itself, due to the secondary circulation, described by ψ, in the horizontal plane [again refer to Eq. (4)]. Therefore, Eq. (18) is the statement that the vertical torque due to the Coriolis force acting on the convective circulation must be in balance with the torque associated with vorticity damping (later dissipation); in the absence of this damping (d1 = 0), there is no balanced circulation. Since we have chosen to model damping linearly, then the response ψ corresponds to a secondary horizontal circulation which is linearly related to the primary poloidal (convective) circulation. That the secondary circulation is singular in the damping coefficient d1 is notable, but not surprising given that equilibrium flow must be in, or nearly in, force balance. Ultimately, in any convective model, it will be the upscale fluxes of momentum, and thermodynamic quantities that are of interest to convective parameterizations, and we will discuss these fluxes in a subsequent manuscript.

We can now eliminate ψ from Eq. (15) using Eq. (18) to arrive at an elliptic equation for the velocity potential in terms of the diabatic heating
∇h2Φ+d1d2N2+d1d2[Φzz+(2Ωd1)2Φnn]=Sz(N2+d1d2).
From Eqs. (19) and (18), along with Eqs. (4) and (5), we can construct all three components of the velocity field from a diabatic heating source. There only involves one elliptic inversion to compute Φ from Eq. (19), a vertical integration of Eq. (18) to compute ψ
ψ=−2Ωd1∫z∞∂Φ∂ndz′,
where the constant of integration is chosen so that the horizontal velocity vanishes at infinite height, and a vertical integration of Eq. (5),
w=∫0z∇h2Φdz′,
where the constant of integration is chosen so that the vertical velocity vanishes at z = 0. Taking the necessary partial derivatives of Φ and ψ in Eq. (4), we have then computed horizontal components of the velocity field.
From the buoyancy equation (3d), we could easily compute b as the deviation of the vertical velocity from WTG, but this expression would be singular in the radiative damping parameter d2 and not illuminating in the WTG limit. Instead, by subtracting the z derivative of the meridional acceleration equation (3c) from the y derivative of the vertical acceleration equation (3d) eliminating ψ using Eq. (18), and performing some antiderivatives, we arrive at the expression for the buoyancy in terms of the velocity potential
b=−d1∫z∞[∇H2Φ+Φzz+(2Ωd1)2Φnn] dz′,
where we have chosen the constant of integration so that the buoyancy vanishes at infinite heights. This equation [Eq. (22)] makes the effect of rotation on buoyancy explicit through the presence of the last term in the integral, and it will be useful when constructing upscale fluxes for convective parameterizations. To determine the pressure perturbation ϕ, we vertically integrate Eq. (3c) using the condition that ϕ vanishes at infinite height
ϕ=∫z∞[d1w−2Ωcos⁡(λ) u−b]dz′.
The exact expression for ϕ in terms of Φ or S is not particularly illuminating, so we leave Eq. (23) as it is. We note, however, that in the absence of buoyancy and damping, Eq. (23) expresses the vertical geostrophic balance discussed by Igel and Biello (2020).

Diffusive momentum damping

Now, we briefly examine the equations when the damping in the momentum equations takes the form of enhanced turbulent diffusivity. Effectively, this corresponds to replacing the momentum drag coefficient with the diffusion operator; d1 → −μ∇2 and every instance of d1 in the denominator should be interpreted as the inversion of the Laplacian. In this case, the equation for the velocity potential becomes
[(N2−μd2∇2)]∇2 ∇h2Φ−d2μ[μ2∇4Φzz+(2Ω)2Φnn]=∇2Sz.
The equations for the other variables follow in much the same manner, and we do not record them here as they do not necessarily provide any more insights into the solutions. However, we note that in the case of diffusive damping, we must solve elliptic equations for all the variables, whereas for linear damping, we need to only solve a single elliptic equation for Φ.

4. The traditional and nontraditional Coriolis terms

We now look at the two cases where either only the NCTs or only the traditional Coriolis terms (TCTs) are retained in Eqs. (3a)–(3e). The former case occurs at the equator and is obtained by setting λ = 0 and ∂/∂n = ∂/∂y. The latter case occurs at the North Pole and is obtained by setting λ = π/2 and ∂/∂n = ∂/∂z. For the purposes of this discussion, instead of using the equation for the velocity potential [Eq. (19)], we will recast it in terms of the vertical velocity by substituting Eq. (5).

Specifically, at the equator, only the nontraditional Coriolis terms are active, and the elliptic equation for the vertical velocity becomes
(N2+d1d2)∇h2w+d1d2(wzz+(2Ωd1)2wyy)=∇h2S,
while the kinematic equation for the streamfunction in terms of the velocity potential becomes
d1∂ψ∂z=2Ω∂Φ∂y.
There are two cases of note that occur at the equator. In the case of d2 = 0, the equation for the vertical velocity is independent of latitude and simplifies to ∇h2w=N−2∇h2S, whose solution is w = N−2S. Thus, in the absence of radiative damping, we obtain the WTG approximation (Hittmeir and Klein 2018), the direct diagnosis of vertical velocity from heating.
The second case occurs if both d1 and d2 are nonzero, but their product is small enough to neglect d1d2, corresponding to d1d2 ≪ N2. In this case, the equation for the vertical velocity at the equator becomes
∇h2w+4Ω2N2d2d1wyy=1N2∇h2S,
which is an equation that would allow for the vertical velocity to be diagnosed directly if not for the term proportional to d2/d1. So in the case of nonzero radiation, we have an equation for the vertical velocity similar to WTG, but with a modification induced by the presence of radiation and the nontraditional Coriolis force terms that requires the inversion of an elliptic operator. Thus, radiation makes the velocity a nonlocal function of the heating, particularly in the meridional direction.

We point out that there are cases that we have not considered where vertical nonlocality induced by the presence of the wzz term in Eq. (25) is important (Kuo and Neelin 2022). Our focus, however, is on the impact of the nontraditional Coriolis terms, which manifest themselves through the wyy term in Eq. (25). By considering the case where d1d2 ≪ N2, we can isolate the impact of the NCTs alone.

Irrespective of the momentum damping coefficient, at the equator, the secondary horizontal circulation described by ψ is proportional to the meridional derivative of Φ—i.e., the horizontal circulation induced by the NCT at the equator is proportional to the meridional component of the poloidal circulation. Thus, we expect poloidal flows which are symmetric about the equator to induce secondary circulations which are antisymmetric about the equator. This symmetry breaking has important implications for upscale momentum fluxes which we will pursue in future work.

At the North Pole, the vertical velocity satisfies
(N2+d1d2)∇h2w+d1d2(1+(2Ωd1)2) wzz=∇h2S,
and the streamfunction is proportional to the velocity potential
d1ψ=2ΩΦ.
The relationship of the streamfunction to the velocity potential in Eq. (29) describes the well-known behavior of geostrophically balanced flows: areas of horizontal convergence of the poloidal flow will drive cyclonic rotation. Usually, this occurs in the lower troposphere where the flow is convergent, while the compensating, divergent, anticyclonic circulation occurs in the upper troposphere.

In contrast to the NCT equation in Eq. (25), where the non-WTG terms (those proportional to d2) manifest as both horizontal and vertical derivatives of w in the elliptic operator, in the case of TCT, given in Eq. (28), the additional term is only proportional to vertical derivatives of w. This wzz term generates a vertically nonlocal response to localized diabatic heating, and it is the effect of damped gravity waves generated by a convective heating source. The coefficient multiplying the vertical derivatives in Eq. (28) is a complicated combination of the rotation rate of Earth, the momentum damping, and the ratio of thermal to momentum damping; that is to say that their effects combine in a manner to be indistinguishable from one another in the solution to the vertical velocity.

In the companion paper, we will present an extensive study of solutions to the balanced framework. But in order to provide a preliminary illustration of the phenomena that the balanced framework describes, we compute approximate solutions for the velocity potential and streamfunction at the equator (NCT) and the North Pole (TCT), for a horizontally localized heating profile, which maximizes in midtroposphere, thereby resembling the latent heat released by a convective cloud,
S={S0N2sin⁡(πz/H),if x2+y2<L0,otherwise,
where S0 = 10−4 m s−3 = 0.36 m s−2 h−1, H = 3 km, L = 3 km, and d1 = 10−4 s−1.

Figure 1a shows a horizontal cross section of the secondary circulation, and the vector field (−ψy, ψx), at the North Pole at the bottom of the troposphere, where only the TCTs are present. In this case, heating drives a cyclonic secondary circulation whose maximum strength occurs at the bottom and top of the troposphere. Figure 1b shows the secondary circulation at the equator in the middle of the troposphere, when only the nontraditional Coriolis terms are present. In this case, the secondary circulation is antisymmetric about the equator. In Fig. 1c, the velocity potential and vector field (−Φx, −Φy) are plotted at the bottom of the troposphere at the North Pole, where only the traditional Coriolis terms are present. Figure 1d shows the velocity potential and vector field (−Φ, −Φy) at the bottom of the troposphere at the equator. In both cases, the flow is convergent at the bottom of the troposphere and divergent at the top.

Fig. 1.
Fig. 1.

(a),(b) Contours of the secondary circulation ψ and the vector field (−ψy, ψx) at the bottom of the troposphere at the North Pole and in the middle of the troposphere at the equator, respectively. (c),(d) Contours of the velocity potential and the vector field (−Φx, −Φy) at the bottom of the troposphere at the North Pole and at the bottom of the troposphere at the equator, respectively. The axes and variables are scaled to the horizontal length scale L, and the color bar is in m s−1.

Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-22-0254.1

5. Summary

In this paper, we discuss a framework for studying convective dynamics under the influence of heating, the full Coriolis force, thermal, and momentum damping. The circulation strengths and length scales we consider allow for the study of steady, linear equilibrated convective flows and constitute the first step in studying momentum and buoyancy fluxes from the convective scales to the mesoscales. We use the Helmholtz decomposition of the velocity field as a tool to disentangle the effects of heating, Coriolis force, and damping on the convective circulation (−Φx, −Φy, w), and the secondary horizontal velocity (−ψy, ψx, 0) that arises in response to it. The schematic panels in Fig. 2 depict the primary convective circulation in the absence of radiative damping and Coriolis force (left), the symmetric primary circulation and the rotational secondary circulation in the presence of radiative damping and the traditional Coriolis force terms (center), and the primary and secondary circulation in the presence of radiative damping and the nontraditional Coriolis force terms (right).

Fig. 2.
Fig. 2.

Schematic representations of the solutions of the convective WTG framework. In purple is the primary poloidal circulation, both the vertical velocity w and the horizontal velocity due to the potential Φ. In yellow is an indication of radiative cooling. Red depicts the secondary circulation ψ, due to the Coriolis force and damping. (a) The WTG without damping or Coriolis force. (b) WTG with radiative cooling and traditional Coriolis terms. (c) WTG with radiative cooling and nontraditional Coriolis terms.

Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-22-0254.1

The framework is encapsulated by two equations. The first equation arises from torque balance, described in Eq. (18), which determines the secondary horizontal circulation ψ, given the velocity potential. The response of the secondary circulation depends on the latitude of the convection; in the absence of the Coriolis terms and radiation, there would only be a poloidal circulation (Fig. 2a). The TCT (poles, Fig. 2b) drives a cyclonic circulation in response to horizontal convergence, while the NCT (equator, Fig. 2c) drives an antisymmetric response proportional to the meridional component of the convective velocity field. The red curves in Figs. 2b and 2c are placed at the heights where the maximum secondary circulation occurs for each case. For the TCT (Fig. 2b), Eq. (29) shows that the secondary circulation is largest at heights where the horizontal convergence of the convection is largest. From Eq. (5), we see that this occurs where the vertical derivative of the vertical velocity (and thus the vertical derivative of the heating) is maximum: at the top and bottom of the troposphere. For the NCT (Fig. 2c), Eq. (26) shows the secondary circulation is largest at the height where the meridional velocity of the convection vanishes. At such elevations, the vertical component of the velocity is maximal; therefore, the secondary circulation due to the NCT is maximal at the height of the maximum upward velocity in the convection.

The second equation in this framework is an elliptic operator [Eq. (19)] whose solution yields the velocity potential Φ, given the diabatic heating S; if dissipation is used instead of drag, then the theory is described by Eq. (24). In the absence of radiative damping, the operator is exactly the weak temperature gradient approximation, but on convective length scales. Radiative damping generates a response in the vertical component of the velocity field away from the diabatic heating source, and thus, we describe this as a nonlocal response. In the case of the NCT [Eq. (28)], the nonlocality is in the vertical direction (Fig. 2c), while in the case of the TCT [Eq. (25)], the nonlocality is in both the vertical and meridional directions (Fig. 2b).

In a companion paper, we study the solutions of this convective–Coriolis balanced framework. Future work will describe the convective momentum and temperature fluxes which arise from diabatic heat sources and the implications of these fluxes for the parameterization of convection in meso- and synoptic-scale dynamics, especially in the tropics.

Acknowledgments.

The authors thank three anonymous reviewers for their helpful comments. This work was partially supported by the NSF under Award AGS-2224293, and by the Krener Assistant Professorship at UC-Davis.

Data availability statement.

No datasets were generated or analyzed during the current study.

REFERENCES

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    • Search Google Scholar
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  • Hirt, M., G. C. Craig, and R. Klein, 2023: Scale interactions between the meso- and synoptic scales and the impact of diabatic heating. Quart. J. Roy. Meteor. Soc., 149, 1319–1334, https://doi.org/10.1002/qj.4456.

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  • Igel, M. R., and J. A. Biello, 2020: The nontraditional Coriolis terms and tropical convective clouds. J. Atmos. Sci., 77, 3985–3998, https://doi.org/10.1175/JAS-D-20-0024.1.

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  • Jeevanjee, N., and D. M. Romps, 2016: Effective buoyancy at the surface and aloft. Quart. J. Roy. Meteor. Soc., 142, 811–820, https://doi.org/10.1002/qj.2683.

    • Search Google Scholar
    • Export Citation

  • Klein, R., 2010: Scale dependent models for atmospheric flows. Annu. Rev. Fluid Mech., 42, 249–274, https://doi.org/10.1146/annurev-fluid-121108-145537.

    • Search Google Scholar
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    • Export Citation

  • Lebovitz, N. R., 1989: The stability equations for rotating, inviscid fluids: Galerkin methods and orthogonal bases. Geophys. Astrophys. Fluid Dyn., 46, 221–243, https://doi.org/10.1080/03091928908208913.

    • Search Google Scholar
    • Export Citation

  • LeMone, M. A., 1983: Momentum transport by a line of cumulonimbus. J. Atmos. Sci., 40, 1815–1834, https://doi.org/10.1175/1520-0469(1983)040<1815:MTBALO>2.0.CO;2.

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  • Manabe, S., and R. F. Strickler, 1964: Thermal equilibrium of the atmosphere with a convective adjustment. J. Atmos. Sci., 21, 361–385, https://doi.org/10.1175/1520-0469(1964)021<0361:TEOTAW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., and F. Schott, 1999: Open-ocean convection: Observations, theory, and models. Rev. Geophys., 37, 1–64, https://doi.org/10.1029/98RG02739.

    • Search Google Scholar
    • Export Citation

  • Ong, H., and P. E. Roundy, 2020: Nontraditional hypsometric equation. Quart. J. Roy. Meteor. Soc., 146, 700–706, https://doi.org/10.1002/qj.3703.

    • Search Google Scholar
    • Export Citation

  • Ong, H., and D. Yang, 2022: The compressional beta effect and convective system propagation. J. Atmos. Sci., 79, 2031–2040, https://doi.org/10.1175/JAS-D-21-0219.1.

    • Search Google Scholar
    • Export Citation

  • Peters, J. M., 2016: The impact of effective buoyancy and dynamic pressure forcing on vertical velocities within two-dimensional updrafts. J. Atmos. Sci., 73, 4531–4551, https://doi.org/10.1175/JAS-D-16-0016.1.

    • Search Google Scholar
    • Export Citation

  • Raymond, D. J., S. L. Sessions, and Ž. Fuchs, 2007: A theory for the spinup of tropical depressions. Quart. J. Roy. Meteor. Soc., 133, 1743–1754, https://doi.org/10.1002/qj.125.

    • Search Google Scholar
    • Export Citation

  • Stewart, A. L., and P. J. Dellar, 2010: Multilayer shallow water equations with complete Coriolis force. Part 1. Derivation on a non-traditional beta-plane. J. Fluid Mech., 651, 387–413, https://doi.org/10.1017/S0022112009993922.

    • Search Google Scholar
    • Export Citation

  • Stewart, A. L., and P. J. Dellar, 2012: Multilayer shallow water equations with complete Coriolis force. Part 2. Linear plane waves. J. Fluid Mech., 690, 16–50, https://doi.org/10.1017/jfm.2011.364.

    • Search Google Scholar
    • Export Citation

  • Stewart, A. L., and P. J. Dellar, 2013: Multilayer shallow water equations with complete Coriolis force. Part 3. Hyperbolicity and stability under shear. J. Fluid Mech., 723, 289–317, https://doi.org/10.1017/jfm.2013.121.

    • Search Google Scholar
    • Export Citation

  • Tort, M., and T. Dubos, 2014: Dynamically consistent shallow-atmosphere equations with a complete Coriolis force. Quart. J. Roy. Meteor. Soc., 140, 2388–2392, https://doi.org/10.1002/qj.2274.

    • Search Google Scholar
    • Export Citation

  • Wing, A. A., K. Emanuel, C. E. Holloway, and C. Muller, 2017: Convective self-aggregation in numerical simulations: A review. Surv. Geophys., 38, 1173–1197, https://doi.org/10.1007/s10712-017-9408-4.

    • Search Google Scholar
    • Export Citation

  • Zhang, K., and G. Schubert, 1997: Linear penetrative spherical rotating convection. J. Atmos. Sci., 54, 2509–2518, https://doi.org/10.1175/1520-0469(1997)054<2509:LPSRC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
Save
Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Adames, Á. F., S. W. Powell, F. Ahmed, V. C. Mayta, and J. D. Neelin, 2021: Tropical precipitation evolution in a buoyancy-budget framework. J. Atmos. Sci., 78, 509–528, https://doi.org/10.1175/JAS-D-20-0074.1.

    • Search Google Scholar
    • Export Citation

  • Bretherton, C. S., and A. H. Sobel, 2002: A simple model of a convectively coupled Walker circulation using the weak temperature gradient approximation. J. Climate, 15, 2907–2920, https://doi.org/10.1175/1520-0442(2002)015<2907:ASMOAC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Chikira, M., 2014: Eastward-propagating intraseasonal oscillation represented by Chikira–Sugiyama cumulus parameterization. Part II: Understanding moisture variation under weak temperature gradient balance. J. Atmos. Sci., 71, 615–639, https://doi.org/10.1175/JAS-D-13-038.1.

    • Search Google Scholar
    • Export Citation

  • Dubos, T., C. Barthlott, and P. Drobinski, 2008: Emergence and secondary instability of Ekman layer rolls. J. Atmos. Sci., 65, 2326–2342, https://doi.org/10.1175/2007JAS2550.1.

    • Search Google Scholar
    • Export Citation

  • Helmholtz, H., 1867: On integrals of the hydrodynamical equations, which express vortex-motion. London Edinburgh Dublin Philos. Mag. J. Sci., 33, 485–512, https://doi.org/10.1080/14786446708639824.

    • Search Google Scholar
    • Export Citation

  • Hirt, M., G. C. Craig, and R. Klein, 2023: Scale interactions between the meso- and synoptic scales and the impact of diabatic heating. Quart. J. Roy. Meteor. Soc., 149, 1319–1334, https://doi.org/10.1002/qj.4456.

    • Search Google Scholar
    • Export Citation

  • Hittmeir, S., and R. Klein, 2018: Asymptotics for moist deep convection I: Refined scalings and self-sustaining updrafts. Theor. Comput. Fluid Dyn., 32, 137–164, https://doi.org/10.1007/s00162-017-0443-z.

    • Search Google Scholar
    • Export Citation

  • Igel, M. R., and J. A. Biello, 2020: The nontraditional Coriolis terms and tropical convective clouds. J. Atmos. Sci., 77, 3985–3998, https://doi.org/10.1175/JAS-D-20-0024.1.

    • Search Google Scholar
    • Export Citation

  • Jeevanjee, N., and D. M. Romps, 2016: Effective buoyancy at the surface and aloft. Quart. J. Roy. Meteor. Soc., 142, 811–820, https://doi.org/10.1002/qj.2683.

    • Search Google Scholar
    • Export Citation

  • Klein, R., 2010: Scale dependent models for atmospheric flows. Annu. Rev. Fluid Mech., 42, 249–274, https://doi.org/10.1146/annurev-fluid-121108-145537.

    • Search Google Scholar
    • Export Citation

  • Kuo, Y.-H., and J. D. Neelin, 2022: Conditions for convective deep inflow. Geophys. Res. Lett., 49, e2022GL100552, https://doi.org/10.1029/2022GL100552.

    • Search Google Scholar
    • Export Citation

  • Lebovitz, N. R., 1989: The stability equations for rotating, inviscid fluids: Galerkin methods and orthogonal bases. Geophys. Astrophys. Fluid Dyn., 46, 221–243, https://doi.org/10.1080/03091928908208913.

    • Search Google Scholar
    • Export Citation

  • LeMone, M. A., 1983: Momentum transport by a line of cumulonimbus. J. Atmos. Sci., 40, 1815–1834, https://doi.org/10.1175/1520-0469(1983)040<1815:MTBALO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Manabe, S., and R. F. Strickler, 1964: Thermal equilibrium of the atmosphere with a convective adjustment. J. Atmos. Sci., 21, 361–385, https://doi.org/10.1175/1520-0469(1964)021<0361:TEOTAW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., and F. Schott, 1999: Open-ocean convection: Observations, theory, and models. Rev. Geophys., 37, 1–64, https://doi.org/10.1029/98RG02739.

    • Search Google Scholar
    • Export Citation

  • Ong, H., and P. E. Roundy, 2020: Nontraditional hypsometric equation. Quart. J. Roy. Meteor. Soc., 146, 700–706, https://doi.org/10.1002/qj.3703.

    • Search Google Scholar
    • Export Citation

  • Ong, H., and D. Yang, 2022: The compressional beta effect and convective system propagation. J. Atmos. Sci., 79, 2031–2040, https://doi.org/10.1175/JAS-D-21-0219.1.

    • Search Google Scholar
    • Export Citation

  • Peters, J. M., 2016: The impact of effective buoyancy and dynamic pressure forcing on vertical velocities within two-dimensional updrafts. J. Atmos. Sci., 73, 4531–4551, https://doi.org/10.1175/JAS-D-16-0016.1.

    • Search Google Scholar
    • Export Citation

  • Raymond, D. J., S. L. Sessions, and Ž. Fuchs, 2007: A theory for the spinup of tropical depressions. Quart. J. Roy. Meteor. Soc., 133, 1743–1754, https://doi.org/10.1002/qj.125.

    • Search Google Scholar
    • Export Citation

  • Stewart, A. L., and P. J. Dellar, 2010: Multilayer shallow water equations with complete Coriolis force. Part 1. Derivation on a non-traditional beta-plane. J. Fluid Mech., 651, 387–413, https://doi.org/10.1017/S0022112009993922.

    • Search Google Scholar
    • Export Citation

  • Stewart, A. L., and P. J. Dellar, 2012: Multilayer shallow water equations with complete Coriolis force. Part 2. Linear plane waves. J. Fluid Mech., 690, 16–50, https://doi.org/10.1017/jfm.2011.364.

    • Search Google Scholar
    • Export Citation

  • Stewart, A. L., and P. J. Dellar, 2013: Multilayer shallow water equations with complete Coriolis force. Part 3. Hyperbolicity and stability under shear. J. Fluid Mech., 723, 289–317, https://doi.org/10.1017/jfm.2013.121.

    • Search Google Scholar
    • Export Citation

  • Tort, M., and T. Dubos, 2014: Dynamically consistent shallow-atmosphere equations with a complete Coriolis force. Quart. J. Roy. Meteor. Soc., 140, 2388–2392, https://doi.org/10.1002/qj.2274.

    • Search Google Scholar
    • Export Citation

  • Wing, A. A., K. Emanuel, C. E. Holloway, and C. Muller, 2017: Convective self-aggregation in numerical simulations: A review. Surv. Geophys., 38, 1173–1197, https://doi.org/10.1007/s10712-017-9408-4.

    • Search Google Scholar
    • Export Citation

  • Zhang, K., and G. Schubert, 1997: Linear penetrative spherical rotating convection. J. Atmos. Sci., 54, 2509–2518, https://doi.org/10.1175/1520-0469(1997)054<2509:LPSRC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    (a),(b) Contours of the secondary circulation ψ and the vector field (−ψy, ψx) at the bottom of the troposphere at the North Pole and in the middle of the troposphere at the equator, respectively. (c),(d) Contours of the velocity potential and the vector field (−Φx, −Φy) at the bottom of the troposphere at the North Pole and at the bottom of the troposphere at the equator, respectively. The axes and variables are scaled to the horizontal length scale L, and the color bar is in m s−1.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Schematic representations of the solutions of the convective WTG framework. In purple is the primary poloidal circulation, both the vertical velocity w and the horizontal velocity due to the potential Φ. In yellow is an indication of radiative cooling. Red depicts the secondary circulation ψ, due to the Coriolis force and damping. (a) The WTG without damping or Coriolis force. (b) WTG with radiative cooling and traditional Coriolis terms. (c) WTG with radiative cooling and nontraditional Coriolis terms.

Fig. 1.

(a),(b) Contours of the secondary circulation ψ and the vector field (−ψy, ψx) at the bottom of the troposphere at the North Pole and in the middle of the troposphere at the equator, respectively. (c),(d) Contours of the velocity potential and the vector field (−Φx, −Φy) at the bottom of the troposphere at the North Pole and at the bottom of the troposphere at the equator, respectively. The axes and variables are scaled to the horizontal length scale L, and the color bar is in m s−1.
Fig. 2.

Schematic representations of the solutions of the convective WTG framework. In purple is the primary poloidal circulation, both the vertical velocity w and the horizontal velocity due to the potential Φ. In yellow is an indication of radiative cooling. Red depicts the secondary circulation ψ, due to the Coriolis force and damping. (a) The WTG without damping or Coriolis force. (b) WTG with radiative cooling and traditional Coriolis terms. (c) WTG with radiative cooling and nontraditional Coriolis terms.
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