Lagrangian transport by vertically confined internal gravity wavepackets

We examine the flows induced by horizontally modulated, vertically confined (or guided), internal wavepackets in a stratified, Boussinesq fluid. The wavepacket induces both an Eulerian flow and a Stokes drift, which together determine the Lagrangian transport of passive tracers. We derive equations describing the wave-induced flows in arbitrary stable stratification and consider four special cases: a two-layer fluid, symmetric and asymmetric piecewise constant (‘top-hat’) stratification and, more representative of the ocean, exponential stratification. In a two-layer fluid, the Stokes drift is positive everywhere with the peak value at the interface, whereas the Eulerian flow is negative and uniform with depth for long groups. Combined, the net depth-integrated Lagrangian transport is zero. If one layer is shallower than the other, the wave-averaged interface displaces into that layer making the Eulerian flow in that layer more negative and the Eulerian flow in the opposite layer more positive so that the depth-integrated Eulerian transports are offset by the same amount in each layer. By contrast, in continuous stratification the depth-integrated transport due to the Stokes drift and Eulerian flow are each zero, but the Eulerian flow is singular if the horizontal phase speed of the induced flow equals the group velocity of the wavepacket, giving rise to a single resonance in uniform stratification (McIntyre, J. Fluid Mech., vol. 60, 1973, pp. 801–811). In top-hat stratification, this single resonance disappears, being replaced by multiple resonances occurring when the horizontal group velocity of the wavepacket matches the horizontal phase speed of higher-order modes. Furthermore, if the stratification is not vertically symmetric, then the Eulerian induced flow varies as the inverse squared horizontal wavenumber for shallow waves, the same as for the asymmetric two-layer case. This ‘infrared catastrophe’ also occurs in the case of exponential stratification suggesting significant backward near-surface transport by the Eulerian induced flow for modulated oceanic internal modes. Numerical simulations are performed confirming these theoretical predictions.

Internal waves in marginally stable abyssal stratified flows

The problem on internal waves in a weakly stratified two-layer fluid is studied semi-analytically. We discuss the 2.5-layer fluid flows with exponential stratification of both layers. The long-wave model describing travelling waves is constructed by means of a scaling procedure with a small Boussinesq parameter. It is demonstrated that solitary-wave regimes can be affected by the Kelvin–Helmholtz instability arising due to interfacial velocity shear in upstream flow.

Internal waves in marginally stable abyssal stratified flows

The problem on internal waves in a weakly stratified two-layered fluid is studied semi-analytically. We discuss the 2.5-layer fluid flows with exponential stratification of both layers. The long-wave model describing travelling waves is constructed by means of scaling procedure with a small Boussinesq parameter. It is demonstrated that solitary wave regimes can be affected by the Kelvin–Helmholtz instability arising due to interfacial velocity shear in upstream flow.

Estimating Subsurface Velocities from Surface Fields with Idealized Stratification

A previously published method by Wang et al. for predicting subsurface velocities and density from sea surface buoyancy and surface height is extended by incorporating analytical solutions to make the vertical projection. One solution employs exponential stratification and the second has a weakly stratified surface layer, approximating a mixed layer. The results are evaluated using fields from a numerical simulation of the North Atlantic. The simple exponential solution yields realistic subsurface density and vorticity fields to nearly 1000 m in depth. Including a mixed layer improves the response in the mixed layer itself and at high latitudes where the mixed layer is deeper. It is in the mixed layer that the surface quasigeostrophic approximation is most applicable. Below that the first baroclinic mode dominates, and that mode is well approximated by the analytical solution with exponential stratification.

Surface Quasigeostrophic Solutions and Baroclinic Modes with Exponential Stratification

The author derives baroclinic modes and surface quasigeostrophic (SQG) solutions with exponential stratification and compares the results to those obtained with constant stratification. The SQG solutions with exponential stratification decay more rapidly in the vertical and have weaker near-surface velocities. This then compounds the previously noted problem that SQG underpredicts the velocities associated with a given surface density anomaly. The author also examines how the SQG solutions project onto the baroclinic modes. With constant stratification, SQG waves larger than deformation scale project primarily onto the barotropic mode and to a lesser degree onto the first baroclinic mode. However, with exponential stratification, the largest projection is on the first baroclinic mode. The effect is even more pronounced over rough bottom topography. Therefore, large-scale SQG waves will look like the first baroclinic mode and vice versa, with realistic stratification.

Nonlinear Internal Waves in Stratified Fluid With Homogeneous Layer

The problem of steady internal waves in a weakly stratified two-layered fluid is studied analytically. We discuss the model with a constant density in lower layer and exponential stratification in the other one. The long-wave approximation using a scaling procedure with small Boussinesq parameter is constructed. The nonlinear ordinary differential equation describing large amplitude solitary waves and internal bores is obtained.

Polar Cap Accretion onto Magnetized Neutron Stars: An Analytic Solution

This abstract should be read in conjunction with the papers by Arons and by Klein and Arons in these proceedings. In the context of the accretion models described there, one can find an analytic solution for the flow down the polar field lines if a number of simplifying assumptions are made. These are (1) steady flow in the co–rotating frame; (2) radiation pressure large compared to gas pressure; (3) pure scattering for the Rosseland opacity, with the magnetic corrections set equal to constants instead of using the actual functions of temperature; (4) diffusion flux of radiative energy proportional to the gradient of the energy density alone, instead of the correct sum of terms proportional to the photon energy density and the number density gradients; and (5) below a radiative shock, subsonic flow in approximate hydrostatic equilibrium. We assumed dipole geometry, and also assume the mass flux is independent of distance from the magnetic axis. The essential trick is to use (1), (2) and (5) to write the advective contribution to the radiation transfer equation as Mg/area = rate at which gravity does work on a fluid element, and use (3) and (4) to write the nonlinear diffusion flux as the ratio of gradients in the energy density. Then the multidimensional diffusion equation can be cast in a separable, linear form by using the logarithmic radial gradient of the energy density as the basic variable (see also Kirk, J., 1985, Astron. and Astrophys., 142, 430). The result is exponential stratification of the energy density, velocity and mass density along B with scale height R*[L(EDeff)/4Lco-p]; the effective Eddington luminosity is discussed by Arons, these proceedings. This result can be understood as the result of almost exact balance between upward diffusion and downward advection of photons in the optically thick medium. The same fluid quantities are stratified in a Gaussian manner across B, with angular half width at half maximum Δθ = [L(EDeff)/Lcap](r/R*)3/2. These distributions agree well with more sophisticated computational results, during times when the flow is steady. When used as a basis for calculations of the radiative entropy, the calculated emergent spectra are not dissimilar to the spectra of high luminosity, accretion powered pulsars.

Forced vertical oscillations in a viscous stratified fluid

The linear fluid dynamics is considered when infinite vertical boundaries are set in oscillatory vertical motion. The case of exponential stratification with constant kinematic viscosity is explicitly studied. When the forcing frequency equals the Brunt–Vaisälä frequency for the fluid, the customary boundary layers are absent in the steady-state oscillation, however small be the kinematic viscosity; for a semi-infinite fluid the corresponding horizontal extent of the region influenced by the boundary motion is then of the order of the stratification length. The sign of the phase angle is everywhere dependent on whether the magnitude of the forcing frequency is greater than or less than that of the Brunt–Vaisäla frequency.