Nontraditional Coriolis Research Articles

Balanced convective circulations in a stratified atmosphere. Part II: circulations in the presence of radiation and the non-traditional coriolis terms

The full Coriolis force consists of terms proportional to the sine and cosine of latitude. The latter, referred to as the non-traditional Coriolis terms, couple the zonal and vertical momentum equations, and are often neglected. When considering the incompressible Euler equations at the equator in the presence of the full Coriolis force, it can be shown that all dynamical fields can be diagnosed from the velocity potential, which itself is the solution to an elliptic equation that depends on latitude, momentum drag, and radiative damping. In this work, we present solutions to these equations for several different types of momentum drag and radiative damping. An important insight of our work is the combined effect that both rotation and radiation have on providing a mechanism for ascent and descent of air away from a localised source of heating.

An unstable mode of the stratified atmosphere under the non-traditional Coriolis acceleration

The traditional approximation neglects the cosine components of the Coriolis acceleration, and this approximation has been widely used in the study of geophysical phenomena. However, the justification of the traditional approximation is questionable under a few circumstances. In particular, dynamics with substantial vertical velocities or geophysical phenomena in the tropics have non-negligible cosine Coriolis terms. Such cases warrant investigations with the non-traditional setting, i.e. the full Coriolis acceleration. In this manuscript, we study the effect of the non-traditional setting on an isothermal, hydrostatic and compressible atmosphere assuming a meridionally homogeneous flow. Employing linear stability analysis, we show that, given appropriate boundary conditions, i.e. a bottom boundary condition that allows for a vertical energy flux and non-reflecting boundary at the top, the atmosphere at rest becomes prone to a novel unstable mode. The validity of assuming a meridionally homogeneous flow is investigated via scale analysis. Numerical experiments were conducted, and Rayleigh damping was used as a numerical approximation for the non-reflecting top boundary. Our three main results are as follows: (i) experiments involving the full Coriolis terms exhibit an exponentially growing instability, yet experiments subjected to the traditional approximation remain stable; (ii) the experimental instability growth rate is close to the theoretical value; (iii) a perturbed version of the unstable mode arises even under sub-optimal bottom boundary conditions. Finally, we conclude our study by discussing the limitations, implications and remaining open questions. Specifically, the influence on numerical deep-atmosphere models and possible physical interpretations of the unstable mode are discussed.

Dynamics of a stratified vortex under the complete Coriolis force: two-dimensional three-components evolution

We study the dynamics of an initially axisymmetric and vertical Lamb–Oseen vortex in a stratified-rotating fluid under the complete Coriolis force on the $f$ plane, i.e. in the presence of a background rotation both along the vertical and horizontal directions. By a combination of direct numerical simulations and asymptotic analyses for small horizontal background rotation, we show that a critical layer appears at the radius where the angular velocity of the vortex is equal to the buoyancy frequency when the Froude number is larger than unity. This critical layer generates a vertical velocity that is invariant along the vertical and which first increases linearly with time and then saturates at an amplitude scaling like $Re^{1/3}/\widetilde{Ro}$ , where $Re$ is the Reynolds number and $\widetilde {Ro}$ the non-traditional Rossby number based on the horizontal component of the background rotation. In turn, a quasi-axisymmetric anomaly of vertical vorticity is produced at the critical radius through the non-traditional Coriolis force. Below a critical $\widetilde {Ro}$ depending on $Re$ , the Rayleigh's inflectional criterion is satisfied and a shear instability is subsequently triggered rendering the vertical vorticity fully non-axisymmetric. The decay of the angular velocity is then enhanced until it is everywhere lower than the buoyancy frequency. A theoretical criterion derived from the Rayleigh condition predicts well the instability. It shows that this phenomenon can occur even for a large non-traditional Rossby number $\widetilde {Ro}$ for large $Re$ . Hence, the non-traditional Coriolis force might have much more impact on geophysical vortices than anticipated by considering the order of magnitude of $\widetilde {Ro}$ .

The Compressional Beta Effect and Convective System Propagation

Abstract The compressional beta effect (CBE) arises in a compressible atmosphere with the nontraditional Coriolis terms (NCTs), the Coriolis force from the locally horizontal part of the planetary rotation. Previous studies proposed that the CBE speeds up the eastward propagation and slows down the westward propagation of zonal vertical circulations in a dry atmosphere. Here, we examine how the CBE affects the propagation of convectively coupled tropical waves. We perform 2D (x, z), large-domain cloud-resolving simulations with and without NCTs. This model setup mimics the atmosphere along Earth’s equator, and differences between the simulations highlight the role of the CBE. We analyze precipitation, precipitable water, and surface and upper-level winds from our simulations. Gravity wave signals emerge in all fields. In the no-NCT simulation, eastward and westward gravity waves propagate at the same speed. With NCTs, eastward gravity waves propagate faster than westward gravity waves. To quantify the strength of the CBE, we then measure the difference in gravity wave speed and find that it linearly increases with the system rotation rate. This result is consistent with our theoretical prediction and suggests that the CBE can induce zonal asymmetry in propagation behaviors of convectively coupled waves. Significance Statement The rotation of Earth turns eastward motion upward and upward motion westward, and vice versa. This effect is called the nontraditional Coriolis effect and is omitted in most of the current atmospheric models for predicting weather and climate. Using an idealized model with cloud physics, this study suggests that the inclusion of the nontraditional Coriolis effect speeds up eastward-moving rainy systems and slows down westward-moving ones. The speed change agrees with a theory without cloud physics. This study encourages restoring the nontraditional Coriolis effect to the atmospheric models since it increases the accuracy of tropical large-scale weather prediction while the cost is low.

Unidirectional Modes Induced by Nontraditional Coriolis Force in Stratified Fluids.

Using topology, we unveil the existence of new unidirectional modes in compressible rotating stratified fluids. We relate their emergence to the breaking of time-reversal symmetry by rotation and vertical mirror symmetry by stratification and gravity. We stress the role of the Coriolis force's nontraditional part, induced by a rotation field tangent to the surface. In contrast with horizontally trapped equatorial waves induced by the traditional component of the Coriolis force perpendicular to the surface, we find vertically trapped modes that propagate along interfaces between regions with distinct stratification properties. We show that such modes are generalized atmospheric Lamb waves whose direction of propagation can be selected by the nontraditional component of the Coriolis force.

Convectively Coupled Equatorial Trapped Waves in Stars and Planets

Abstract In this paper, we have studied convectively coupled equatorially trapped waves in rotating stars, with and without magnetic fields. The equatorial trapped hydrodynamic and magnetohydrodynamic Poincaré, Rossby, mixed Rossby–Poincaré, and Kelvin waves were identified. The effects of stratification and nontraditional Coriolis force terms have been investigated. When the flow is strongly stratified, the wave frequencies of the convectively coupled model are almost the same as those of a shallow-water model. However, when the flow is weakly stratified, the wave frequencies are constrained by the buoyancy frequency. The nontraditional Coriolis terms affect the widths and phases of the equatorial waves. The width increases with an increasing nontraditional Coriolis parameter. Phase shift occurs when the nontraditional Coriolis parameter is included. The magnetic effect is significant when the magnetic field is strong. We have applied the model in the solar atmosphere and solar tachocline to explain the Rieger-type periodicities. For the solar atmosphere, when the magnetic effect is taken into account, we find that the magnetic field should be smaller than 5 G in the solar photosphere. Otherwise, the Rieger-type periodicities can only be attributed to long Rossby waves. For the solar tachocline, we find that magnetic field of the solar tachocline should be smaller than 50 kG to observe the 160 day Rieger period. In addition, we find that the effect of the nontraditional Coriolis terms is not obvious in the solar photosphere, but its effect on the tachocline is significant.

The Nontraditional Coriolis Terms and Tropical Convective Clouds

AbstractThe full, three-dimensional Coriolis force includes the familiar sine-of-latitude terms as well as frequently dropped cosine-of-latitude terms [nontraditional Coriolis terms (NCT)]. The latter are often ignored because they couple the zonal and vertical momentum equations that in the large-scale limit of weak vertical velocity are considered insignificant almost everywhere. Here, we ask whether equatorial mesoscale clouds that fall outside the large-scale limit are affected by the NCT. A simple scaling indicates that a Lagrangian parcel convecting at 10 m s−1 through the depth of the troposphere should be deflected over 2 km to the west. To understand the real impact of NCT, we develop a mathematical framework that describes an azimuthally symmetric convective circulation with an analytical expression for an incompressible poloidal flow. Because the model incorporates the full three-dimensional flow associated with convection, it uniquely predicts not only the westward tilt of clouds but also a meridional diffluence of western cloud outflow. To test these predictions, we perform a set of cloud-resolving simulations whose results show preferential lifting of surface parcels with positive zonal momentum and zonal asymmetry in convective strength. RCE simulations show changes to the organization of coherent precipitation regions and a decrease in mean convective intensity of approximately 2 m s−1 above the freezing level. An additional pair of dry cloud-resolving simulations designed to mimic the steady-state flow of the model show maximum perturbations to the upper-level zonal flow of 8 m s−1. Together, the numerical and analytic results suggest the NCT consequentially alter equatorial mesoscale convective circulations and should be considered in conceptual models.

The Compressional Beta Effect: Analytical Solution, Numerical Benchmark, and Data Analysis

AbstractThis study derives a complete set of equatorially confined wave solutions from an anelastic equation set with the complete Coriolis terms, which include both the vertical and meridional planetary vorticity. The propagation mechanism can change with the effective static stability. When the effective static stability reduces to neutral, buoyancy ceases, but the role of buoyancy as an eastward-propagation mechanism is replaced by the compressional beta effect (i.e., vertical density-weighted advection of the meridional planetary vorticity). For example, the Kelvin mode becomes a compressional Rossby mode. Compressional Rossby waves are meridional vorticity disturbances that propagate eastward owing to the compressional beta effect. The compressional Rossby wave solutions can serve as a benchmark to validate the implementation of the nontraditional Coriolis terms (NCTs) in numerical models; with an effectively neutral condition and initial large-scale disturbances given a half vertical wavelength spanning the troposphere on Earth, compressional Rossby waves are expected to propagate eastward at a phase speed of 0.24 m s−1. The phase speed increases with the planetary rotation rate and the vertical wavelength and also changes with the density scale height. Besides, the compressional beta effect and the meridional vorticity tendency are reconstructed using reanalysis data and regressed upon tropical precipitation filtered for the Madden–Julian oscillation (MJO). The results suggest that the compressional beta effect contributes 10.8% of the meridional vorticity tendency associated with the MJO in terms of the ratio of the minimum values.

Nontraditional hypsometric equation

AbstractThis study corrects the hypsometric equation by restoring the nontraditional terms to relax the hydrostatic approximation. The nontraditional terms include one Coriolis term and two metric terms in the vertical momentum equation. The hypsometric equations with and without the nontraditional correction are used to calculate the geopotential height of pressure levels using more than 300,000 selected tropical rawinsonde profiles. With westerlies between two pressure levels, the thickness of the layer increases, which reduces the upward pressure gradient forces to balance the upward nontraditional Coriolis forces; the opposite is true for easterlies. Hence, zonal winds are negatively correlated with traditional geopotential height biases aloft. At 500 hPa, for example, traditional geopotential height error in the tropical troposphere is on the order of at least 0.5 m, which is considerable with respect to geopotential height variability in tropical large‐scale flow, on the order of 10 to 15 m.

Linear effects of nontraditional Coriolis terms on intertropical convergence zone forced large‐scale flow

AbstractThis article promotes a measure to validate the hydrostatic approximation via scaling the nontraditional Coriolis term (NCT) in the zonal momentum equation. To demonstrate the scaling, this study simulates large‐scale flow forced by a prescribed heat source, mimicking the intertropical convergence zone (ITCZ) using a linearized forced‐dissipative model. The model solves two similar equations, between which the only difference is the inclusion of NCTs. The equations are derived using the following approximations: anelastic, equatorial beta‐plane, linearized, zonally symmetric, steady, and a constant dissipation coefficient. The large‐scale flows simulated with and without NCTs are compared in terms of the meridional–vertical circulation, the zonal wind, and the potential temperature. Both results appear like the Hadley circulation. With the model parameters controlled, the differences between the results without NCTs and those with NCTs yield the linear biases due to omitting NCTs. The most prominent bias is a westerly wind bias in the ITCZ heating region that emerges because omitting NCTs prevents the associated westward acceleration when heating‐induced vertical motion is present. The zonal wind bias divided by the zonal wind with NCTs is 0.120 ± 0.007 in terms of the westerly maximum and 0.0452 ± 0.0005 in terms of the root mean square (RMS) when the prescribed ITCZ mimics the observed ITCZ in May over the East Pacific. These normalized measures of the zonal wind bias increase with a narrower ITCZ or an ITCZ closer to the Equator, because of a weaker subtropical jet stream given the same vertical heating profile. This difference can be traced by a nondimensional parameter scaling the ratio of the NCT to the traditional Coriolis term. The scaling encourages the restoration of NCTs into global models.

Poleward propagation of near-inertial waves induced by fluctuating winds over a baroclinically unstable zonal jet

We investigate the excitation and radiation of near-inertial internal gravity waves continuously excited by a latitudinally confined temporally fluctuating wind in a numerical model of a stratified ocean on a $\unicode[STIX]{x1D6FD}$-plane at mid-latitude. The surface wind forcing contains both high- and low-frequency components which excite propagating waves and a baroclinically unstable zonal jet respectively. Wentzel–Kramers–Brillouin (WKB) ray theory implies that near-inertial waves propagate strictly towards the equator. We seek to refine this view here by (i) adding the non-traditional Coriolis force (accounting for the horizontal component of the Earth’s rotation) into the equations of motion, in order to allow poleward sub-inertial propagation to occur, and (ii) relaxing the conceptual constraint of no zonal variability, to allow the zonal jet to undergo instability, to meander and to sustain an active field of mesoscale eddies, potentially impacting the excitation of near-inertial waves. The key results are that, while (i) permits weakly stratified waveguides with sub-inertial poleward wave propagation to develop in accord with theory, the sub-inertial energy flux observed is very small compared with the equatorward flux. Thus, in terms of energy radiated from the storm track, non-traditional effects are small for wind-driven near-inertial waves. The consequences of (ii) are much more pronounced. Refinement (ii) produces a radiating wave field that is bidirectional, i.e. with both poleward and equatorward components. We show that the presence of regions of significant background vorticity with horizontal scales significantly smaller than the width of the storm track provides the scale selection mechanism to excite waves with sufficiently super-inertial frequencies to propagate poleward distances of the order of 1000 km.

Some properties of tidal currents estimated from analytical and LES simulation studies

Some features of tidal current ellipses near the ocean floor are estimated from the bottom-limit solutions of the tidal equations with constant eddy viscosity. It is verified that a clockwise rotating ellipse becomes broad descending through the bottom boundary, whereas a counterclockwise ellipse becomes narrow except near the critical latitude. The major axis of an ellipse near the floor is directed 45° leftward (rightward) against that of the overlying tidal flow ellipse poleward from the critical latitude in the northern (southern) hemisphere. Equatorward from the critical latitude, the major axis near the floor lines up with that of the overlying tidal current ellipse in both hemispheres. The retrograde rotating hodograph against the overlying clockwise tidal hodograph appears near the floor around the critical latitude. These features are confirmed from the numerical studies using the large eddy simulation model. The deflection of the major axis poleward from the critical latitude is a little smaller, about 19°. Furthermore, the effects of nontraditional Coriolis force resulting from the horizontal component of the earth-rotating vector on the development of turbulence were investigated. When the diurnal tidal current is directed westward, the effective vertical eddy viscosity becomes maximum with developing turbulence. While in the case of a semidiurnal tide, a northward tidal current is preferred for turbulence development.

The Importance of the Nontraditional Coriolis Terms in Large-Scale Motions in the Tropics Forced by Prescribed Cumulus Heating

AbstractIn meteorological dynamics, the shallow-atmosphere approximation is generally used in the momentum equation, together with the “traditional approximation.” In the traditional approximation, two cosine Coriolis terms [hereafter called nontraditional Coriolis terms (NCTs)] and some metric terms are omitted. However, some studies have suggested that the omission of the NCTs may not be appropriate for conditions near the equator. Therefore, this paper investigates the effect of the NCTs on large-scale motions forced by local positive-only heating mimicking cumulus convection.The authors use the linearized quasihydrostatic equation system on an equatorial β plane. Prescribed heating is assumed to move eastward with the slow phase speed of an intraseasonal period. Results of scale analysis and numerical calculations show that differences with and without the NCTs (hereafter, contributions) exhibit the following four features. (i) Whereas contributions to horizontal divergence and vertical velocity are small, the NCTs have large effects on horizontal velocity (vertical vorticity) and perturbations of pressure, potential temperature, and density. (ii) Contributions to horizontal velocity and pressure perturbation have equivalent barotropic structure. (iii) Contributions show east–west asymmetric patterns, with large contributions appearing at the western side of the forcing. (iv) Contributions to vertical vorticity and pressure perturbation are extremely large when the meridional gradient of heating is large. These features can be comprehensively explained by the tilting of the meridional component of the planetary vorticity, which is caused by the meridional gradient of heating.

Multilayer shallow water equations with complete Coriolis force. Part 2. Linear plane waves

AbstractWe investigate the behaviour of linear plane waves in multilayer shallow water equations that include a complete treatment of the Coriolis force. These equations improve upon the conventional shallow water equations, based on the traditional approximation, that include only the part of the Coriolis force due to the locally vertical component of the rotation vector. Including the complete Coriolis force leads to dramatic changes in the structure of long linear plane waves. It allows subinertial waves to exist with frequencies below the inertial frequency, the minimum frequency for which waves exist under the traditional approximation. These subinertial waves are characterized by a distinguished limit in which the horizontal pressure gradient becomes comparable to the upwellings and downwellings driven by the non-traditional Coriolis term in the vertical momentum equation. The subinertial waves connect wave modes that remain separate in the conventional multilayer shallow water equations, such as the surface and internal waves in a two-layer system. Eastward-propagating surface waves in a two-layer system connect with westward-propagating internal waves, and vice versa, via the long subinertial waves. The long subinertial waves cannot be classified as either surface or internal waves, due to the phase difference between the disturbances to the interfaces in these waves.

Equatorial Zonal Jet Formation through the Barotropic Instability of Low-Frequency Mixed Rossby–Gravity Waves, Equilibration by Inertial Instability, and Transition to Superrotation

Abstract Depth-dependent barotropic instability of short mixed Rossby–gravity (MRG) waves is proposed as a mechanism for the formation of equatorial zonal jets. High-resolution primitive equation simulations show that a single MRG wave of very short zonal wavelength and small to moderate amplitude is unstable and leads to the development of a largely barotropic, zonally symmetric flow, featuring a westward jet at the equator and extra-equatorial jets alternating in direction with latitude. At higher but still moderate amplitude, westward flow still prevails at the equator at depths of maximum horizontal velocity amplitude in the initial wave, but the long-term equilibrated state can also feature eastward “superrotating” jets at the equator near the depths of zero horizontal velocity in the initial wave. The formation of the superrotating jets in the simulations is found to be sensitive to the inclusion of the nontraditional Coriolis force in the equations of motion. A linear theory is used to demonstrate the existence of exponentially growing horizontally nondivergent perturbations with a dominant zonally symmetric zonal velocity component. An argument for the sense and positioning of the jets relative to the equator is given in terms of inertial instability and the meridional mixing of planetary vorticity by the small zonal-scale components of the linearly unstable modes. In the long time evolution of the flow, if the amplitude of the westward equatorial jet becomes too great, zonally symmetric inertial instability limits the growth of the jets, and inertial adjustment leads to the homogenization of potential vorticity in latitude and depth around the equator.

A mechanism of deep-ocean mixing due to near-inertial waves generated by flow over bottom topography

This article describes a remarkable effect of the horizontal component of the Coriolis vector, which is traditionally neglected in geophysical fluid wave theory, for the generation of inertio-gravity waves caused by flow over topography. Earlier an initial-value approach was used to calculate the evolutions of internal waves generated by atmospheric forcing in the linear Boussinesq model as done by many investigators for the study of near-inertial waves in the ocean. In this study, however, we focused on the mechanism of generating near-inertial waves by a bottom forcing with various periods. It is shown that the non-traditional Coriolis effects can significantly enhance the generation of near-inertial waves. Moreover, the intensity of generated near-inertial waves increases as the period of bottom forcing approaches to the local inertial period, suggesting the importance of tidal flows for generating near-inertial waves in deep ocean near critical latitudes. It is inferred that such near-inertial waves may contribute to a mechanism of deep-ocean mixing.

Symmetric Stability of Compressible Zonal Flows on a Generalized Equatorial β Plane

Abstract Sufficient conditions are derived for the linear stability with respect to zonally symmetric perturbations of a steady zonal solution to the nonhydrostatic compressible Euler equations on an equatorial β plane, including a leading order representation of the Coriolis force terms due to the poleward component of the planetary rotation vector. A version of the energy–Casimir method of stability proof is applied: an invariant functional of the Euler equations linearized about the equilibrium zonal flow is found, and positive definiteness of the functional is shown to imply linear stability of the equilibrium. It is shown that an equilibrium is stable if the potential vorticity has the same sign as latitude and the Rayleigh centrifugal stability condition that absolute angular momentum increase toward the equator on surfaces of constant pressure is satisfied. The result generalizes earlier results for hydrostatic and incompressible systems and for systems that do not account for the nontraditional Coriolis force terms. The stability of particular equilibrium zonal velocity, entropy, and density fields is assessed. A notable case in which the effect of the nontraditional Coriolis force is decisive is the instability of an angular momentum profile that decreases away from the equator but is flatter than quadratic in latitude, despite its satisfying both the centrifugal and convective stability conditions.