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We extend the Matsuno–Gill model, originally developed on the equatorial $\\beta$-plane, to the surface of the sphere. While on the $\\beta$-plane the non-dimensional model contains a single parameter, the damping rate $\\gamma$, on a sphere the model contains a second parameter, the rotation rate $\\epsilon ^{1/2}$ (Lamb number). By considering the different combinations of damping and rotation, we are able to characterize the solutions over the $(\\gamma, \\epsilon ^{1/2})$ plane. We find that the $\\beta$-plane approximation is accurate only for fast rotation rates, where gravity waves traverse a fraction of the sphere's diameter in one rotation period. The particular solutions studied by Matsuno and Gill are accurate only for fast rotation and moderate damping rates, where the relaxation time is comparable to the time on which gravity waves traverse the sphere's diameter. Other regions of the parameter space can be described by different approximations, including radiative relaxation, geostrophic, weak temperature gradient and non-rotating approximations. The effect of the additional parameter introduced by the sphere is to alter the eigenmodes of the free system. Thus, unlike the solutions obtained by Matsuno and Gill, where the long-term response to a symmetric forcing consists solely of Kelvin and Rossby waves, the response on the sphere includes other waves as well, depending on the combination of $\\gamma$ and $\\epsilon ^{1/2}$. The particular solutions studied by Matsuno and Gill apply to Earth's oceans, while the more general $\\beta$-plane solutions are only somewhat relevant to Earth's troposphere. In Earth's stratosphere, Venus and Titan, only the spherical solutions apply.

The Cretaceous ‘greenhouse’ period (~145 to ~66 million years ago, Ma) in Earth’s history is relatively well documented by multiple paleoproxy records, which indicate that the meridional sea surface temperature (SST) gradient increased (non-monotonically) from the Valanginian (~135 Ma) to the Maastrichtian (~68 Ma). Changes in atmospheric CO 2 concentration, solar constant, and paleogeography are the primary drivers of variations in the spatiotemporal distribution of SST. However, the particular contribution of each of these drivers (and their underlying mechanisms) to changes in the SST distribution remains poorly understood. Here we use data from a suite of paleoclimate simulations to compare the relative effects of atmospheric CO 2 variability and paleogeographic changes on mid-latitudinal SST gradient through the Cretaceous. Further, we use a fundamental model of wind-driven ocean gyres to quantify how changes in the Northern Hemisphere paleogeography weaken the circulation in subtropical ocean gyres, leading to an increase in extratropical SSTs. Simulations using a coupled atmosphere-ocean model show that paleogeography-driven reduction in the intensity of surface ocean circulation explains much of the increase in the mid-latitudinal sea surface temperature gradient during the Cretaceous.

An intermediate complexity moist general circulation model is used to investigate the sensitivity of the quasi‐biennial oscillation (QBO) to resolution, diffusion, tropical tropospheric waves, and parameterized gravity waves. Finer horizontal resolution is shown to lead to a shorter period, while finer vertical resolution is shown to lead to a longer period and to a larger amplitude in the lowermost stratosphere. More scale‐selective diffusion leads to a faster and stronger QBO, while enhancing the sources of tropospheric stationary wave activity leads to a weaker QBO. In terms of parameterized gravity waves, broadening the spectral width of the source function leads to a longer period and a stronger amplitude although the amplitude effect saturates in the mid‐stratosphere when the half‐width exceeds ∼25m/s. A stronger gravity wave source stress leads to a faster and stronger QBO, and a higher gravity wave launch level leads to a stronger QBO. All of these sensitivities are shown to result from their impact on the resultant wave‐driven momentum torque in the tropical stratosphere. Atmospheric models have struggled to accurately represent the QBO, particularly at moderate resolutions ideal for long climate integrations. In particular, capturing the amplitude and penetration of QBO anomalies into the lower stratosphere (which has been shown to be critical for the tropospheric impacts) has proven a challenge. The results provide a recipe to generate and/or improve the simulation of the QBO in an atmospheric model. Plain Language Summary The most prominent mode of variability in the tropical stratosphere is the quasi‐biennial oscillation (QBO), characterized by easterly and westerly winds alternating sign every ∼14 months. Only relatively recently have comprehensive models begun to simulate a QBO spontaneously, and even in these models the representation of the QBO typically suffers from biases. Here we elucidate the sensitivities of the QBO to a wide range of model parameters, and explore how these parameters affect the QBO behavior. We expect that these results will be helpful for improving the QBO in more comprehensive models. Key Points Sensitivity of the quasi‐biennial oscillation (QBO) to resolution, dissipation, wave forcing, and parameterized gravity waves is explored in a single framework The influence of these factors on the QBO can be related to their impact on wave‐induced momentum fluxes in the deep tropics The QBO period can be tuned independently of its amplitude, but the vertical structure (particularly at lower levels) is harder to capture

Simple analytic models developed in this study are applied to long‐term averages of reanalysis surface salinity data to quantify two fundamental properties of ocean currents. The first model is based on the new Freshening Length schema and its application to the Irminger Current yields a ratio of about 5 between the turbulent entrainment rates of surrounding fresher surface waters west and east of Greenland. The second model is based on the steady solution of the advection‐diffusion equation subject to suitable boundary conditions. The application of this model to the spreading of fresh, snow‐melt, water from the delta of the Po river in the northwest Adriatic Sea into the rest of the Sea yields a ratio of 8 × 104 m between the eddy exchange coefficient and the speed of advection in the Sea. Plain Language Summary Differences in ocean water salinity were used for over a century to quantify the horizontal fluxes in and out of evaporative, motionless, basins such as the Mediterranean Sea. In the present study we develop simple expressions based on analytic models that extend the century‐old approach to ocean currents where the water is constantly moving rather than remaining stagnant. The models developed here are combined with long‐term data of sea surface salinity along two currents—the salty Irminger Current that flows around the southern tip of Greenland and the flow of fresh snow‐melt water from the Po river into the Adriatic Sea. The models and climatological data used here yield quantitative estimates of two basic parameters: (a) the rate at which a high‐salinity current detrains salt to the surrounding ocean. (b) The balance between the slow downstream propagation and eddy (turbulent) exchange coefficient. The models developed in this study can be applied to other currents and regions of the world ocean. Key Points In some oceanic circumstances, changes in Sea Surface Salinity (SSS) gradient provide a simple, reliable and robust diagnostic of ocean currents Changes in the entrainment rate of surrounding water into a current, correspond to observable changes in SSS gradient Reanalysis SSS data quantify the ratio between the speed of advection and the eddy exchange coefficient in slow currents

The equivalent depth of an atmospheric layer is of importance in determining the phase speed of gravity waves and characterizing wave phenomena. The value of the equivalent depth can be obtained from the eigenvalues of the vertical structure equation (the vertical part of the primitive equations) where the mean temperature profile is a coefficient. Both numerical solutions of the vertical structure equation and analytical considerations are employed to calculate the equivalent depth, hn, as a function of the atmospheric layer's thickness, Δz. Our solutions for layers of thickness 100 ≤Δz≤ 2000 m show that for baroclinic modes, hn can be over two orders of magnitudes smaller than Δz. Analytic expressions are derived for hn in layers of uniform temperature and numerical solutions are derived for layers in which the temperature changes linearly with height. A comparison between the two cases shows that a slight temperature gradient (of say 0.65 K across a 100 m layer) decreases hn by a factor of 3 (but can reach a factor of 10 for larger gradients) compared with its value in a layer of uniform temperature, while a change of 10 K in the layer's uniform temperature hardly changes hn. The n=0 baroclinic mode exists in all combinations of boundary conditions top and bottom while the barotropic mode only exists when the vertical velocity vanishes at both boundaries of the layer. Solutions of the eigenvalue problem associated with the vertical structure equation show that the equivalent depth of an atmospheric layer, hn, can be O(100) smaller than the layer's thickness, Δz. The combination of boundary conditions on L, the eigenfunction of the vertical structure equation, and/or dLdz strongly affects the modes. In layers where the temperature decreases linearly with height the equivalent depth can be smaller by one order of magnitude compared with layers in which the temperature is assumed uniform.

The seminal Ekman (1905) f-plane theory of wind-driven transport at the ocean surface is extended to the β plane by substituting the pseudo-angular momentum for the zonal velocity in the Lagrangian equation. When the β term is added, the equations become nonlinear, which greatly complicates the analysis. Though rotation relates the momentum equations in the zonal and the meridional directions, the transformation to pseudo-angular momentum greatly simplifies the longitudinal dynamics, which yields a clear description of the meridional dynamics in terms of a slow drift compounded by fast oscillations; this can then be applied to describe the motion in the zonal direction. Both analytical expressions and numerical calculations highlight the critical role of the Equator in determining the trajectories of water columns forced by eastward-directed (in the Northern Hemisphere) wind stress even when the water columns are initiated far from the Equator. Our results demonstrate that the averaged motion in the zonal direction depends on the amplitude of the meridional oscillations and is independent of the direction of the wind stress. The zonal drift is determined by a balance between the initial conditions and the magnitude of the wind stress, so it can be as large as the mean meridional motion; i.e., the averaged flow direction is not necessarily perpendicular to the wind direction.

A time series of the 17O excess (17Δ) was measured in the Gulf of Elat (Aqaba) between May 2007 and August 2009. 17Δ is unaffected by respiration; thus it is a unique, conservative tracer that preserves the signature acquired in the photic zone, the source region for deep-water formation. In this study we used 17Δ to assess the ratio of photosynthetic O₂ to atmospheric O₂ in deep water. We observed an increase of 17Δ by 20 per meg in the water residing below 300 m over a period of 3 months, followed by a decrease of 60 per meg over the next 7 months. These changes indicated penetration of photosynthetic O₂, followed by penetration of atmospheric O₂ into the deep water. To test whether vertical mixing could explain the observed variations in 17Δ, we compared our results with simulated values obtained from a one-dimensional hydrodynamic model, which was extended to include dissolved O₂ isotopes. Although successfully reproducing the observed temperatures, salinities, and dissolved O₂ concentrations in the gulf, the model could not reproduce the observed variations in 17Δ in the deep water. This discrepancy shows that horizontal mixing processes have an important role in the interaction between deep and surface water in the gulf. We suggest that for the most part, these processes occur along the coastal boundaries of the gulf.

Low-frequency waves that develop in a shallow layer of fluid, contained in a channel with linearly slopping bottom and rotating with uniform angular speed are investigated theoretically and experimentally. Exact numerical solutions of the eigenvalue problem, obtained from the linearized shallow water equations on the f-plane, show that the waves are trapped near the channel's shallow wall and propagate along it with the shallow side on their right in the Northern hemisphere. The phase speed of the waves is slower compared with that of the harmonic theory in which bottom slope is treated inconsistently. A first-order approximation of the cross-channel dependence of the coefficient in the eigenvalue equation yields an approximation of the cross-channel velocity eigenfunction as an Airy function, which, for sufficiently wide channels, yields an explicit expression for the wave's dispersion relation. The analytic solutions of the eigenvalue problem agree with the numerical solutions in both the wave trapping and the reduced phase speed. For narrow channels, our theory yields an estimate of the channel width below which the harmonic theory provides a more accurate approximation. Laboratory experiments were conducted on a 13 m diameter turntable at LEGI-Coriolis (France) into which a linearly sloping bottom of 10 % incline was installed. A wavemaker generated waves of known frequency at one end of the turntable and the wavenumbers of these waves were measured at the opposite end using a particle imaging velocimetry technique. The experimental results regarding the phase speed and the radial structure of the amplitude are in very good agreement with our theoretical non-harmonic predictions, which support the present modification of the harmonic theory in wide channels.

The applicability of one-dimensional (zonally invariant) harmonic and trapped wave theories for Inertia-Gravity waves to simulations on the mid-latitude β-plane is examined by comparing the analytical estimates in the geostrophic adjustment and Ekman adjustment problems with numerical simulations of the linearized rotating shallow water equations. The spatial average of the absolute differences between the theoretical solutions and the simulations, ϵ(t), is calculated for values of the domain's north-south extent, L, ranging from L=4 to L=60 (where L is measured in units of the deformation radius). The comparisons show that: (i) though ϵ oscillates with time, its low-pass filter, ϵ.sup.LP (t), increases with time. (ii) In small domains, ϵ.sup.LP (t) in harmonic theory is significantly smaller than in trapped wave theory, while the opposite occurs in large domains. (iii) The accuracy of the harmonic wave theory decreases with L for 0L20, while for L20 the trend changes with time. (iv) The accuracy of the trapped wave theory increases with L in the geostrophic adjustment problem, while in the Ekman adjustment problem, its best accuracy is obtained when Lâ30. (v) There is a range of L and t values for which no theory provides reasonable approximations, and this range is wider in the Ekman adjustment problem than in the geostrophic adjustment problem. Non-linear simulations of a multilayered stratified ocean show that internal inertia-gravity waves exhibit the same characteristics as the wave solutions of the linearized rotating shallow water equations in a single layer ocean.

In this work, trajectories of more than 500 drogued surface drifters launched in the equatorial ocean since 1979 are analyzed by employing the results of a new Lagrangian theory of poleward transport from the Equator forced by the prevailing trade winds. The Lagrangian theory provides an explicit expression for the depth of the Ekman layer that circumvents the application of the 3D continuity equation that requires calculation of the divergence of horizontal transport, which has been the basis of all previous studies on the subject. The analysis is carried out for drifters launched within 1° of the Equator that reached a final latitude of 3, 4, or 5° north or south of the Equator while also remaining in one hemisphere throughout their entire travel time. The analysis yields robust estimates of 45 m for the Ekman layer's depth and 1.0 m d−1 for the upwelling speed of thermocline water into the layer.