Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 1. pseudomomentum-based theory

There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics. In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms. The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent. This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir & Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii). In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.

DK-1D: a drift-kinetic simulation tool for modelling the shear Alfvén wave and its interaction with collisionless plasma

We present a highly accurate tool for the simulation of shear Alfven waves (SAW) in collisionless plasma. SAW are important in space plasma environments because for small perpendicular scale lengths they can support an electric field parallel to the ambient magnetic field. Electrons can be accelerated by the parallel electric field and these waves have been implicated as the source of vibrant auroral displays. However, the parallel electric field carried by SAW is small in comparison to the perpendicular electric field of the wave, making it difficult to measure directly in the laboratory, or by satellites in the near-Earth plasma environment. In this paper, we present a simulation code that provides a means to study in detail the SAW-particle interaction in both space and laboratory plasma. Using idealised, small-amplitude propagating waves with a single perpendicular wavenumber, the simulation code accurately reproduces the damping rates and parallel electric field amplitudes predicted by linear theory for varying temperatures and perpendicular scale lengths. We present a rigorous kinetic derivation of the parallel electric field strength for small-amplitude SAW and show that commonly-used inertial and kinetic approximations are valid except for where the ratio of thermal to Alfv\'{e}n speed is between 0.7 and 1.0. We also present nonlinear simulations of large-amplitude waves and show that in cases of strong damping, the damping rates and parallel electric field strength deviate from linear predictions when wave energies are greater than only a few percent of the plasma kinetic energy, a situation which is often observed in the magnetosphere. The drift-kinetic code provides reliable, testable predictions of the parallel electric field strength which can be investigated directly in the laboratory, and will help to bridge the gap between studies of SAW in man-made and naturally occuring plasma.

Nonlinear wave-activity conservation laws and Hamiltonian structure for the two-dimensional anelastic equations

Exact, finite-amplitude, local wave-activity conservation laws are derived for disturbances to steady flows in the context of the two-dimensional anelastic equations. The conservation laws are expressed entirely in terms of Eulerian quantities, and have the property that, in the limit of a small-amplitude, slowly varying, monochromatic wave train, the wave-activity density A and flux F, when averaged over phase, satisfy F = cgA where cg is the group velocity of the waves. For nonparallel steady flows, the only conserved wave activity is a form of disturbance pseudoenergy; when the steady flow is parallel, there is in addition a conservation law for the disturbance pseudomomentum. The above results are obtained not only for isentropic background states (which give the so-called “deep form” of the anelastic equations), but also for arbitrary background potential-temperature profiles θ0(z) so long as the variation in θ0(z) over the depth of the fluid is small compared with θ0 itself. The Hamiltonian structure of the equations is established in both cases, and its symmetry properties discussed. An expression for available potential energy is also derived that, for the case of a stably stratified background state (i.e., dθ0/dz > 0), is locally positive definite; the expression is valid for fully three-dimensional flow. The counterparts to results for the two-dimensional Boussinesq equations are also noted.

Projection of global wave climate change towards the end of the 21st century

Wind generated waves at the sea surface are of outstanding importance for both their practical relevance in many aspects, such as coastal erosion, protection, or safety of navigation, and for their scientific relevance in modifying fluxes at the air-sea interface. So far long-term changes in ocean wave climate have been studied mostly from a regional perspective with global dynamical studies emerging only recently. Here a global wave climate study is presented, in which a global wave model (WAM) is driven by atmospheric forcing from a global climate model (ECHAM5) for present day and potential future climate conditions represented by the IPCC (Intergovernmental Panel for Climate Change) A1B emission scenario. It is found that changes in mean and extreme wave climate towards the end of the twenty-first century are small to moderate, with the largest signals being a poleward shift in the annual mean and extreme significant wave heights in the mid-latitudes of both hemispheres, more pronounced in the Southern Hemisphere, and most likely associated with a corresponding shift in mid-latitude storm tracks. These changes are broadly consistent with results from the few studies available so far. The projected changes in the mean wave periods, associated with the changes in the wave climate in the mid to high latitudes, are also shown, revealing a moderate increase in the equatorial eastern side of the ocean basins. This study presents a step forward towards a larger ensemble of global wave climate projections required to better assess robustness and uncertainty of potential future wave climate change.

Runge-Kutta IMEX schemes for the Horizontally Explicit/Vertically Implicit (HEVI) solution of wave equations

Many operational weather forecasting centres use semi-implicit time-stepping schemes because of their good efficiency. However, as computers become ever more parallel, horizontally explicit solutions of the equations of atmospheric motion might become an attractive alternative due to the additional inter-processor communication of implicit methods. Implicit and explicit (IMEX) time-stepping schemes have long been combined in models of the atmosphere using semi-implicit, split-explicit or HEVI splitting. However, most studies of the accuracy and stability of IMEX schemes have been limited to the parabolic case of advection–diffusion equations. We demonstrate how a number of Runge–Kutta IMEX schemes can be used to solve hyperbolic wave equations either semi-implicitly or HEVI. A new form of HEVI splitting is proposed, UfPreb, which dramatically improves accuracy and stability of simulations of gravity waves in stratified flow. As a consequence it is found that there are HEVI schemes that do not lose accuracy in comparison to semi-implicit ones. The stability limits of a number of variations of trapezoidal implicit and some Runge–Kutta IMEX schemes are found and the schemes are tested on two vertical slice cases using the compressible Boussinesq equations split into various combinations of implicit and explicit terms. Some of the Runge–Kutta schemes are found to be beneficial over trapezoidal, especially since they damp high frequencies without dropping to first-order accuracy. We test schemes that are not formally accurate for stiff systems but in stiff limits (nearly incompressible) and find that they can perform well. The scheme ARK2(2,3,2) performs the best in the tests.

Wave activity in the tropical tropopause layer in seven reanalysis and four chemistry climate model data sets

Sub-seasonal variability including equatorial waves significantly influence the dehydration and transport processes in the tropical tropopause layer (TTL). This study investigates the wave activity in the TTL in 7 reanalysis data sets (RAs; NCEP1, NCEP2, ERA40, ERA-Interim, JRA25, MERRA, and CFSR) and 4 chemistry climate models (CCMs; CCSRNIES, CMAM, MRI, and WACCM) using the zonal wave number-frequency spectral analysis method with equatorially symmetric-antisymmetric decomposition. Analyses are made for temperature and horizontal winds at 100 hPa in the RAs and CCMs and for outgoing longwave radiation (OLR), which is a proxy for convective activity that generates tropopause-level disturbances, in satellite data and the CCMs. Particular focus is placed on equatorial Kelvin waves, mixed Rossby-gravity (MRG) waves, and the Madden-Julian Oscillation (MJO). The wave activity is defined as the variance, i.e., the power spectral density integrated in a particular zonal wave number-frequency region. It is found that the TTL wave activities show significant difference among the RAs, ranging from ∼0.7 (for NCEP1 and NCEP2) to ∼1.4 (for ERA-Interim, MERRA, and CFSR) with respect to the averages from the RAs. The TTL activities in the CCMs lie generally within the range of those in the RAs, with a few exceptions. However, the spectral features in OLR for all the CCMs are very different from those in the observations, and the OLR wave activities are too low for CCSRNIES, CMAM, and MRI. It is concluded that the broad range of wave activity found in the different RAs decreases our confidence in their validity and in particular their value for validation of CCM performance in the TTL, thereby limiting our quantitative understanding of the dehydration and transport processes in the TTL.

The gravity wave momentum flux in hydrostatic flow with directional shear over elliptical mountains

Semi-analytical expressions for the momentum flux associated with orographic internal gravity waves, and closed analytical expressions for its divergence, are derived for inviscid, stationary, hydrostatic, directionally-sheared flow over mountains with an elliptical horizontal cross-section. These calculations, obtained using linear theory conjugated with a third-order WKB approximation, are valid for relatively slowly-varying, but otherwise generic wind profiles, and given in a form that is straightforward to implement in drag parametrization schemes. When normalized by the surface drag in the absence of shear, a quantity that is calculated routinely in existing drag parametrizations, the momentum flux becomes independent of the detailed shape of the orography. Unlike linear theory in the Ri → ∞ limit, the present calculations account for shear-induced amplification or reduction of the surface drag, and partial absorption of the wave momentum flux at critical levels. Profiles of the normalized momentum fluxes obtained using this model and a linear numerical model without the WKB approximation are evaluated and compared for two idealized wind profiles with directional shear, for different Richardson numbers (Ri). Agreement is found to be excellent for the first wind profile (where one of the wind components varies linearly) down to Ri = 0.5, while not so satisfactory, but still showing a large improvement relative to the Ri → ∞ limit, for the second wind profile (where the wind turns with height at a constant rate keeping a constant magnitude). These results are complementary, in the Ri > O(1) parameter range, to Broad’s generalization of the Eliassen–Palm theorem to 3D flow. They should contribute to improve drag parametrizations used in global weather and climate prediction models.

Self-consistent wave-particle interactions in dispersive scale long-period field-line-resonances

Using 1D Vlasov drift-kinetic computer simulations, it is shown that electron trapping in long period standing shear Alfven waves (SAWs) provides an efficient energy sink for wave energy that is much more effective than Landau damping. It is also suggested that the plasma environment of low altitude auroral-zone geomagnetic field lines is more suited to electron acceleration by inertial or kinetic scale Alfven waves. This is due to the self-consistent response of the electron distribution function to SAWs, which must accommodate the low altitude large-scale current system in standing waves. We characterize these effects in terms of the relative magnitude of the wave phase and electron thermal velocities. While particle trapping is shown to be significant across a wide range of plasma temperatures and wave frequencies, we find that electron beam formation in long period waves is more effective in relatively cold plasma.

Systematic model forecast error in Rossby wave structure

Diabatic processes can alter Rossby wave structure; consequently errors arising from model processes propagate downstream. However, the chaotic spread of forecasts from initial condition uncertainty renders it difficult to trace back from root mean square forecast errors to model errors. Here diagnostics unaffected by phase errors are used, enabling investigation of systematic errors in Rossby waves in winter-season forecasts from three operational centers. Tropopause sharpness adjacent to ridges decreases with forecast lead time. It depends strongly on model resolution, even though models are examined on a common grid. Rossby wave amplitude reduces with lead time up to about five days, consistent with under-representation of diabatic modification and transport of air from the lower troposphere into upper-tropospheric ridges, and with too weak humidity gradients across the tropopause. However, amplitude also decreases when resolution is decreased. Further work is necessary to isolate the contribution from errors in the representation of diabatic processes.

The Madden-Julian Oscillation and the diurnal cycle over the Maritime Continent: scale interactions and modelling

The Maritime Continent archipelago, situated on the equator at 95-165E, has the strongest land-based precipitation on Earth. The latent heat release associated with the rainfall affects the atmospheric circulation throughout the tropics and into the extra-tropics. The greatest source of variability in precipitation is the diurnal cycle. The archipelago is within the convective region of the Madden-Julian Oscillation (MJO), which provides the greatest variability on intra-seasonal time scales: large-scale (∼10^7 km^2) active and suppressed convective envelopes propagate slowly (∼5 m s^-1) eastwards between the Indian and Pacific Oceans. High-resolution satellite data show that a strong diurnal cycle is triggered to the east of the advancing MJO envelope, leading the active MJO by one-eighth of an MJO cycle (∼6 days). Where the diurnal cycle is strong its modulation accounts for 81% of the variability in MJO precipitation. Over land this determines the structure of the diagnosed MJO. This is consistent with the equatorial wave dynamics in existing theories of MJO propagation. The MJO also affects the speed of gravity waves propagating offshore from the Maritime Continent islands. This is largely consistent with changes in static stability during the MJO cycle. The MJO and its interaction with the diurnal cycle are investigated in HiGEM, a high-resolution coupled model. Unlike many models, HiGEM represents the MJO well with eastward-propagating variability on intra-seasonal time scales at the correct zonal wavenumber, although the inter-tropical convergence zone's precipitation peaks strongly at the wrong time, interrupting the MJO's spatial structure. However, the modelled diurnal cycle is too weak and its phase is too early over land. The modulation of the diurnal amplitude by the MJO is also too weak and accounts for only 51% of the variability in MJO precipitation. Implications for forecasting and possible causes of the model errors are discussed, and further modelling studies are proposed.

The physics of orographic gravity wave drag

The drag and momentum fluxes produced by gravity waves generated in flow over orography are reviewed, focusing on adiabatic conditions without phase transitions or radiation effects, and steady mean incoming flow. The orographic gravity wave drag is first introduced in its simplest possible form, for inviscid, linearized, non-rotating flow with the Boussinesq and hydrostatic approximations, and constant wind and static stability. Subsequently, the contributions made by previous authors (primarily using theory and numerical simulations) to elucidate how the drag is affected by additional physical processes are surveyed. These include the effect of orography anisotropy, vertical wind shear, total and partial critical levels, vertical wave reflection and resonance, non-hydrostatic effects and trapped lee waves, rotation and nonlinearity. Frictional and boundary layer effects are also briefly mentioned. A better understanding of all of these aspects is important for guiding the improvement of drag parametrization schemes.

Compensation between resolved wave driving and parameterized orographic gravity wave driving of the Brewer–Dobson circulation and its response to climate change

Following recent findings, the interaction between resolved (Rossby) wave drag and parameterized orographic gravity wave drag (OGWD) is investigated, in terms of their driving of the Brewer–Dobson circulation (BDC), in a comprehensive climate model. To this end, the parameter that effectively determines the strength of OGWD in present-day and doubled CO2 simulations is varied. The authors focus on the Northern Hemisphere during winter when the largest response of the BDC to climate change is predicted to occur. It is found that increases in OGWD are to a remarkable degree compensated by a reduction in midlatitude resolved wave drag, thereby reducing the impact of changes in OGWD on the BDC. This compensation is also found for the response to climate change: changes in the OGWD contribution to the BDC response to climate change are compensated by opposite changes in the resolved wave drag contribution to the BDC response to climate change, thereby reducing the impact of changes in OGWD on the BDC response to climate change. By contrast, compensation does not occur at northern high latitudes, where resolved wave driving and the associated downwelling increase with increasing OGWD, both for the present-day climate and the response to climate change. These findings raise confidence in the credibility of climate model projections of the strengthened BDC.

Impact of non-hydrostatic effects and trapped lee waves on mountain wave drag in directionally sheared flow

The orographic gravity wave drag produced in flow over an axisymmetric mountain when both vertical wind shear and non-hydrostatic effects are important was calculated using a semi-analytical two-layer linear model, including unidirectional or directional constant wind shear in a layer near the surface, above which the wind is constant. The drag behaviour is determined by partial wave reflection at the shear discontinuity, wave absorption at critical levels (both of which exist in hydrostatic flow), and total wave reflection at levels where the waves become evanescent (an intrinsically non-hydrostatic effect), which produces resonant trapped lee wave modes. As a result of constructive or destructive wave interference, the drag oscillates with the thickness of the constant-shear layer and the Richardson number within it (Ri), generally decreasing at low Ri and when the flow is strongly non-hydrostatic. Critical level absorption, which increases with the angle spanned by the wind velocity in the constant-shear layer, shields the surface from reflected waves, keeping the drag closer to its hydrostatic limit. While, for the parameter range considered here, the drag seldom exceeds this limit, a substantial drag fraction may be produced by trapped lee waves, particularly when the flow is strongly non-hydrostatic, the lower layer is thick and Ri is relatively high. In directionally sheared flows with Ri = O(1), the drag may be misaligned with the surface wind in a direction opposite to the shear, a behaviour which is totally due to non-trapped waves. The trapped lee wave drag, whose reaction force on the atmosphere is felt at low levels, may therefore have a distinctly different direction from the drag associated with vertically propagating waves, which acts on the atmosphere at higher levels.

Analysis of wave phenomena in a morphogenetic mechanochemical model and an application to post-fertilization waves on eggs

Wave solutions to a mechanochemical model for cytoskeletal activity are studied and the results applied to the waves of chemical and mechanical activity that sweep over an egg shortly after fertilization. The model takes into account the calcium-controlled presence of actively contractile units in the cytoplasm, and consists of a viscoelastic force equilibrium equation and a conservation equation for calcium. Using piecewise linear caricatures, we obtain analytic solutions for travelling waves on a strip and demonstrate uiat the full nonlinear system behaves as predicted by the analytic solutions. The equations are solved on a sphere and the numerical results are similar to the analytic solutions. We indicate how the speed of the waves can be used as a diagnostic tool with which the chemical reactivity of the egg surface can be measured.