Abstract. The analytic wave-solutions obtained by Matsuno (1966) in his seminal work on equatorial waves provide a simple and informative way of assessing atmospheric and oceanic models by measuring the accuracy with which they simulate these waves. These solutions approximate the solutions of the shallow water equations on the sphere for small speeds of gravity waves such as those of the baroclinic modes in the atmosphere and ocean. This is in contrast to the solutions of the non-divergent barotropic vorticity equation, used in the Rossby-Haurwitz test case, which are only accurate for large speeds of gravity waves such as those of the barotropic mode. The proposed test case assigns specific values to the wave-parameters (gravity wave speed, zonal wave-number, meridional wave-mode and amplitude) for both planetary and inertia gravity waves, and confirms the accuracy of the simulation by employing Hovmöller diagrams and temporal and spatial spectra. The proposed test case is successfully applied to a standard finite-difference, equatorial, non-linear, shallow water model in spherical coordinates, which demonstrates that Matsuno’s wave-solutions can be accurately simulated for at least 10 wave-periods, which for oceanic planetary waves is nearly 1300 days. In order to facilitate the use of the proposed test case, we provide Matlab, Python and Fortran codes for computing the analytic solutions at any time on arbitrary latitude-longitude grids.

Abstract. The analytic wave solutions obtained by Matsuno (1966) in his seminal work on equatorial waves provide a simple and informative way of assessing the performance of atmospheric models by measuring the accuracy with which they simulate these waves. These solutions approximate the solutions of the shallow-water equations on the sphere for low gravity-wave speeds such as those of the baroclinic modes in the atmosphere. This is in contrast to the solutions of the non-divergent barotropic vorticity equation, used in the Rossby–Haurwitz test case, which are only accurate for high gravity-wave speeds such as those of the barotropic mode. The proposed test case assigns specific values to the wave parameters (gravity-wave speed, zonal wave number, meridional wave mode and wave amplitude) for both planetary and inertia-gravity waves, and suggests simple assessment criteria suitable for zonally propagating wave solutions. The test is successfully applied to a spherical shallow-water model in an equatorial channel and to a global-scale model. By adding a small perturbation to the initial fields it is demonstrated that the chosen initial waves remain stable for at least 100 wave periods. The proposed test case can also be used as a resolution convergence test.

We study the fundamental process of geostrophic adjustment in infinitely long zonal and meridional channels of widths Ly and Lx, respectively, by deriving analytic solutions and simulating the linearized rotating shallow water equations (LRSWE). All LRSWE's variables are divided into time-independent (geostrophic) and time-dependent components. The latter includes Kelvin waves, Poincaré waves, and inertial oscillations. Explicit expressions are derived for both components, which are confirmed numerically. Anti-symmetric and symmetric initial height distributions, η0(x), are considered, both of which introduce a length scale, D, into the problem. We show that for an anti-symmetric η0(x), (i) the rate of approach to geostrophy is D-independent; (ii) the decay rate of inertial oscillations is ∝t−1/2; and (iii) for DLy≫1, the energy of the final state in any finite sub-domain of the channel exceeds that of the initial state, while for DLy≪1 the energy in the final state is smaller than in the initial state. In contrast, for a symmetric η0(x): (i) the rate of approach to geostrophy increases with D; (ii) the decay rate of inertial oscillations is ∝t−3/2, and (iii) the energy of the final state is always smaller than that of the initial state. In meridional channels, the effect of boundaries is to (i) block the waves, propagation to infinity; (ii) alter the spatial structure of the geostrophic flow; and (iii) discretize the frequency spectrum, thus eliminating the inertial oscillations. The ratio of wave energy to initial energy decreases with Lx for anti-symmetric η0(x) and increases with Lx for symmetric η0(x).

<p>The theory of the transition from an unbalanced initial state to a geostrophically balanced state, referred to as geostrophic adjustment, is a fundamental theory in geophysical fluid dynamics. The theory originated in the 1930s on the f-plane and since then the theory was barely advanced to the β-plane. The present study partially fills the gap by extending the geostrophic adjustment theory to the β-plane in the case of resting fluid with a step-like initial height distribution η<sub>0</sub>. In the presentation, we focus on the one-dimensional adjustment theory in a zonally-invariant, finite, meridional domain of width L where η<sub>0</sub> =<em> </em>η<sub>0</sub>(y). By solving the linearized rotating shallow water equations numerically, the effect of β on the adjustment process is examined primarily from the wave perspective while the spatial structure of the geostrophic steady-state will be addressed only briefly. The gradient of η<sub>0</sub>(y) is aligned perpendicular to the domain walls in our zonally-invariant set-up which implies that the geostrophic state only represents the time-averaged solution over many wave periods rather than a steady-state that is reached by the system at long times. We found that: (i) the effect of β on the geostrophic state is significant only for b = cot(φ<sub>0</sub>)R<sub>d</sub>/R ≥ 0.5 (where R<sub>d</sub> is the radius of deformation, R is Earth's radius and φ<sub>0</sub> is the central latitude of the domain). (ii) In wide domains the effect of β on the waves is significant even for small b (e.g. b=0.005). EOF analysis demonstrates that for b=0.005 and in narrow domains (e.g. L = 4R<sub>d</sub>) harmonic wave theory provides an accurate approximation for the waves, while in wide domains (e.g. L = 60R<sub>d</sub>) accurate approximations are provided by the trapped wave theory. Preliminary results derived in the two-dimensional case, where η<sub>0</sub> =<em> </em>η<sub>0</sub>(x) is symmetric, imply that the results outlined in item (ii) above hold in this case too. </p>

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.