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.
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.
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.
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.
Diffusive momentum damping
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).
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.
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.
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.
(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).
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.
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