Generation of inertia-gravity waves in the rotating thermal annulus by a localised boundary layer instability

Waves with periods shorter than the inertial period exist in the atmosphere (as inertia-gravity waves) and in the oceans (as Poincaré and internal gravity waves). Such waves owe their origin to various mechanisms, but of particular interest are those arising either from local secondary instabilities or spontaneous emission due to loss of balance. These phenomena have been studied in the laboratory, both in the mechanically-forced and the thermally-forced rotating annulus. Their generation mechanisms, especially in the latter system, have not yet been fully understood, however. Here we examine short period waves in a numerical model of the rotating thermal annulus, and show how the results are consistent with those from earlier laboratory experiments. We then show how these waves are consistent with being inertia-gravity waves generated by a localised instability within the thermal boundary layer, the location of which is determined by regions of strong shear and downwelling at certain points within a large-scale baroclinic wave flow. The resulting instability launches small-scale inertia-gravity waves into the geostrophic interior of the flow. Their behaviour is captured in fully nonlinear numerical simulations in a finite-difference, 3D Boussinesq Navier-Stokes model. Such a mechanism has many similarities with those responsible for launching small- and meso-scale inertia-gravity waves in the atmosphere from fronts and local convection.

The equatorial response to higher latitude forcing

The classic view, following Charney and Webster and Holton, is that significant midlatitude forcing of the Tropics can be expected only in regions with westerly winds in the upper troposphere because it is only in these regions that stationary Rossby waves will be able to propagate toward the equator. Here it is shown that higherlatitude forcing can project directly onto equatorial waves and give a significant tropical response in both easterly and westerly tropical flow. The equatorial response to higher-latitude forcing is considered in the context of a dry atmosphere and a localized higher-latitude forcing with eastward or westward phase speed. Previous ideas of the Doppler shifting of equatorial waves by zonal flows are extended to include consideration of a forcing involving a range of zonal wavenumbers. A Gill-type model suggests that there can be significant forcing of equatorial waves by either vorticity forcing or heating in higher latitudes. In agreement with the theory, the Kelvin wave response to eastward forcing is peaked at high frequencies/short periods but reduces only slowly with decreasing frequency. Primitive-equation experiments confirm the strong equatorial response associated with a deep Kelvin wave for forcing in midlatitudes. The response is strongest in the Eastern Hemisphere with its equatorial, upper-tropospheric easterlies. The possible importance of this equatorial response in the organization of large-scale, deep tropical convection and the initiation of the Madden–Julian oscillation is discussed. The ability of westward forcing in higher latitudes to trigger Rossby–gravity and Rossby waves is found in the primitive-equation model to be significant but rather less robust. These wave signatures are clearest in the lower troposphere. For shorter periods the Rossby–gravity wave dominates, and for upper-tropospheric forcing, downward and eastward wave activity propagation is seen. Upper-tropospheric westerlies are found to enhance the response.

Equivalence of weak formulations of the steady water waves equations

We prove the equivalence of three weak formulations of the steady water waves equations, namely: the velocity formulation, the stream function formulation and the Dubreil-Jacotin formulation, under weak Hölder regularity assumptions on their solutions.

The interaction of flexural-gravity waves with periodic geometries

A periodic structure of finite extent is embedded within an otherwise uniform two-dimensional system consisting of finite-depth fluid covered by a thin elastic plate. An incident harmonic flexural-gravity wave is scattered by the structure. By using an approximation to the corresponding linearised boundary value problem that is based on a slowly varying structure in conjunction with a transfer matrix formulation, a method is developed that generates the whole solution from that for just one cycle of the structure, providing both computational savings and insight into the scattering process. Numerical results show that variations in the plate produce strong resonances about the ‘Bragg frequencies’ for relatively few periods. We find that certain geometrical variations in the plate generate these resonances above the Bragg value, whereas other geometries produce the resonance below the Bragg value. The familiar resonances due to periodic bed undulations tend to be damped by the plate.

Relaxing the Tropics to an ‘observed’ state: analysis using a simple baroclinic model

The technique of relaxation of the tropical atmosphere towards an analysis in a month-season forecast model has previously been successfully exploited in a number of contexts. Here it is shown that when tropical relaxation is used to investigate the possible origin of the observed anomalies in June–July 2007, a simple dynamical model is able to reproduce the observed component of the pattern of anomalies given by an ensemble of ECMWF forecast runs. Following this result, the simple model is used for a range of experiments on time-scales of relaxation, variables and regions relaxed based on a control model run with equatorial heating in a zonal flow. A theory based on scale analysis for the large-scale tropics is used to interpret the results. Typical relationships between scales are determined from the basic equations, and for a specified diabatic heating a chain of deductions for determining the dependent variables is derived. Different critical time-scales are found for tropical relaxation of different dependent variables to be effective. Vorticity has the longest critical time-scale, typically 1.2 days. For temperature and divergence, the time-scales are 10 hours and 3 hours, respectively. However not all the tropical fields, in particular the vertical motion, are reproduced correctly by the model unless divergence is heavily damped. To obtain the correct extra-tropical fields, it is crucial to have the correct rotational flow in the subtropics to initiate the Rossby wave propagation from there. It is sufficient to relax vorticity or temperature on a time-scale comparable or less than their critical time-scales to obtain this. However if the divergent advection of vorticity is important in the Rossby Wave Source then strong relaxation of divergence is required to accurately represent the tropical forcing of Rossby waves.

The Stokes conjecture for waves with vorticity

We study stagnation points of two-dimensional steady gravity free-surface water waves with vorticity. We obtain for example that, in the case where the free surface is an injective curve, the asymptotics at any stagnation point is given either by the “Stokes corner flow” where the free surface has a corner of 120°, or the free surface ends in a horizontal cusp, or the free surface is horizontally flat at the stagnation point. The cusp case is a new feature in the case with vorticity, and it is not possible in the absence of vorticity. In a second main result we exclude horizontally flat singularities in the case that the vorticity is 0 on the free surface. Here the vorticity may have infinitely many sign changes accumulating at the free surface, which makes this case particularly difficult and explains why it has been almost untouched by research so far. Our results are based on calculations in the original variables and do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity.

Vertical propagation of information in a middle atmosphere data assimilation system by gravity-wave drag feedbacks

The mesospheric response to the 2002 Antarctic Stratospheric Sudden Warming (SSW) is analysed using the Canadian Middle Atmosphere Model Data Assimilation System (CMAM-DAS), where it represents a vertical propagation of information from the observations into the data-free mesosphere. The CMAM-DAS simulates a cooling in the lowest part of the mesosphere which is accomplished by resolved motions, but which is extended to the mid- to upper mesosphere by the response of the model's non-orographic gravity-wave drag parameterization to the change in zonal winds. The basic mechanism is that elucidated by Holton consisting of a net eastward wave-drag anomaly in the mesosphere during the SSW, although in this case there is a net upwelling in the polar mesosphere. Since the zonal-mean mesospheric response is shown to be predictable, this demonstrates that variations in the mesospheric state can be slaved to the lower atmosphere through gravity-wave drag.

A linear model for the structure of turbulence beneath surface water waves

The structure of turbulence in the ocean surface layer is investigated using a simplified semi-analytical model based on rapid-distortion theory. In this model, which is linear with respect to the turbulence, the flow comprises a mean Eulerian shear current, the Stokes drift of an irrotational surface wave, which accounts for the irreversible effect of the waves on the turbulence, and the turbulence itself, whose time evolution is calculated. By analysing the equations of motion used in the model, which are linearised versions of the Craik–Leibovich equations containing a ‘vortex force’, it is found that a flow including mean shear and a Stokes drift is formally equivalent to a flow including mean shear and rotation. In particular, Craik and Leibovich’s condition for the linear instability of the first kind of flow is equivalent to Bradshaw’s condition for the linear instability of the second. However, the present study goes beyond linear stability analyses by considering flow disturbances of finite amplitude, which allows calculating turbulence statistics and addressing cases where the linear stability is neutral. Results from the model show that the turbulence displays a structure with a continuous variation of the anisotropy and elongation, ranging from streaky structures, for distortion by shear only, to streamwise vortices resembling Langmuir circulations, for distortion by Stokes drift only. The TKE grows faster for distortion by a shear and a Stokes drift gradient with the same sign (a situation relevant to wind waves), but the turbulence is more isotropic in that case (which is linearly unstable to Langmuir circulations).

On the momentum fluxes associated with mountain waves in directionally sheared flows

The direct impact of mountain waves on the atmospheric circulation is due to the deposition of wave momentum at critical levels, or levels where the waves break. The first process is treated analytically in this study within the framework of linear theory. The variation of the momentum flux with height is investigated for relatively large shears, extending the authors’ previous calculations of the surface gravity wave drag to the whole atmosphere. A Wentzel–Kramers–Brillouin (WKB) approximation is used to treat inviscid, steady, nonrotating, hydrostatic flow with directional shear over a circular mesoscale mountain, for generic wind profiles. This approximation must be extended to third order to obtain momentum flux expressions that are accurate to second order. Since the momentum flux only varies because of wave filtering by critical levels, the application of contour integration techniques enables it to be expressed in terms of simple 1D integrals. On the other hand, the momentum flux divergence (which corresponds to the force on the atmosphere that must be represented in gravity wave drag parameterizations) is given in closed analytical form. The momentum flux expressions are tested for idealized wind profiles, where they become a function of the Richardson number (Ri). These expressions tend, for high Ri, to results by previous authors, where wind profile effects on the surface drag were neglected and critical levels acted as perfect absorbers. The linear results are compared with linear and nonlinear numerical simulations, showing a considerable improvement upon corresponding results derived for higher Ri.

Estimation of the friction velocity in stably stratified boundary-layer flows over hills

A method is suggested for the calculation of the friction velocity for stable turbulent boundary-layer flow over hills. The method is tested using a continuous upstream mean velocity profile compatible with the propagation of gravity waves, and is incorporated into the linear model of Hunt, Leibovich and Richards with the modification proposed by Hunt, Richards and Brighton to include the effects of stability, and the reformulated solution of Weng for the near-surface region. Those theoretical results are compared with results from simulations using a non-hydrostatic microscale-mesoscale two-dimensional numerical model, and with field observations for different values of stability. These comparisons show a considerable improvement in the behaviour of the theoretical model when the friction velocity is calculated using the method proposed here, leading to a consistent variation of the boundary-layer structure with stability, and better agreement with observational and numerical data.

Mountain waves in two-layer sheared flows: Critical-level effects, wave reflection, and drag enhancement

Internal gravity waves generated in two-layer stratified shear flows over mountains are investigated here using linear theory and numerical simulations. The impact on the gravity wave drag of wind profiles with constant unidirectional or directional shear up to a certain height and zero shear above, with and without critical levels, is evaluated. This kind of wind profile, which is more realistic than the constant shear extending indefinitely assumed in many analytical studies, leads to important modifications in the drag behavior due to wave reflection at the shear discontinuity and wave filtering by critical levels. In inviscid, nonrotating, and hydrostatic conditions, linear theory predicts that the drag behaves asymmetrically for backward and forward shear flows. These differences primarily depend on the fraction of wavenumbers that pass through their critical level before they are reflected by the shear discontinuity. If this fraction is large, the drag variation is not too different from that predicted for an unbounded shear layer, while if it is small the differences are marked, with the drag being enhanced by a considerable factor at low Richardson numbers (Ri). The drag may be further enhanced by nonlinear processes, but its qualitative variation for relatively low Ri is essentially unchanged. However, nonlinear processes seem to interact constructively with shear, so that the drag for a noninfinite but relatively high Ri is considerably larger than the drag without any shear at all.

Drag produced by trapped lee waves and propagating mountain waves in a two-layer atmosphere

The surface drag force produced by trapped lee waves and upward propagating waves in non-hydrostatic stratified flow over a mountain ridge is explicitly calculated using linear theory for a two-layer atmosphere with piecewise-constant static stability and wind speed profiles. The behaviour of the drag normalized by its hydrostatic single-layer reference value is investigated as a function of the ratio of the Scorer parameters in the two layers l_2/l_1 and of the corresponding dimensionless interface height l_1 H, for selected values of the dimensionless ridge width l_1 a and ratio of wind speeds in the two layers. When l_2/l_1 → 1, the propagating wave drag approaches 1 in approximately hydrostatic conditions, and the trapped lee wave drag vanishes. As l_2/l_1 decreases, the propagating wave drag progressively displays an oscillatory behaviour with l_1 H, with maxima of increasing magnitude due to constructive interference of reflected waves in the lower layer. The trapped lee wave drag shows localized maxima associated with each resonant trapped lee wave mode, occurring for small l_2/l_1 and slightly higher values of l_1 H than the propagating wave drag maxima. As l1a decreases, i.e. the flow becomes more non-hydrostatic, the propagating wave drag decreases and the regions of non-zero trapped lee wave drag extend to higher l_2/l_1. These results are confirmed by numerical simulations for l_2/l_1 = 0.2. In parameter ranges of meteorological relevance, the trapped lee wave drag may have a magnitude comparable to that of propagating wave drag, and be larger than the reference single-layer drag. This may have implications for drag parametrization in global climate and weather-prediction models.

Multispacecraft observation of magnetic cloud erosion by magnetic reconnection during propagation

During propagation, Magnetic Clouds (MC) interact with their environment and, in particular, may reconnect with the solar wind around it, eroding away part of its initial magnetic flux. Here we quantitatively analyze such an interaction using combined, multipoint observations of the same MC flux rope by STEREO A, B, ACE, WIND and THEMIS on November 19–20, 2007. Observation of azimuthal magnetic flux imbalance inside a MC flux rope has been argued to stem from erosion due to magnetic reconnection at its front boundary. The present study adds to such analysis a large set of signatures expected from this erosion process. (1) Comparison of azimuthal flux imbalance for the same MC at widely separated points precludes the crossing of the MC leg as a source of bias in flux imbalance estimates. (2) The use of different methods, associated errors and parametric analyses show that only an unexpectedly large error in MC axis orientation could explain the azimuthal flux imbalance. (3) Reconnection signatures are observed at the MC front at all spacecraft, consistent with an ongoing erosion process. (4) Signatures in suprathermal electrons suggest that the trailing part of the MC has a different large-scale magnetic topology, as expected. The azimuthal magnetic flux erosion estimated at ACE and STEREO A corresponds respectively to 44% and 49% of the inferred initial azimuthal magnetic flux before MC erosion upon propagation. The corresponding average reconnection rate during transit is estimated to be in the range 0.12–0.22 mV/m, suggesting most of the erosion occurs in the inner parts of the heliosphere. Future studies ought to quantify the influence of such an erosion process on geo-effectiveness.