Diffusive Staircases in Shear: Dynamics and Heat Transport

Arctic staircases mediate the heat transport from the warm water of Atlantic origin to the cooler waters of the Arctic mixed layer. For this reason, staircases have received much due attention from the community, and their heat transport has been well characterized for systems in the absence of external forcing. However, the ocean is a dynamic environment with large-scale currents and internal waves being omnipresent, even in regions shielded by sea-ice. Thus, we have attempted to address the effects of background shear on fully developed staircases using numerical simulations. The code, which is pseudo-spectral, evolves the governing equations for a Boussinesq fluid with temperature and salinity in a shearing coordinate system. We find that—– unlike many other double-diffusive systems—the sheared staircase requires three-dimensional simulations to properly capture the dynamics. Our simulations predict shear patterns that are consistent with observations and show that staircases in the presence of external shear should be expected to transport heat and salt at least twice as efficiently as in the corresponding non-sheared systems. These findings may lead to critical improvements in the representation of micro-scale mixing in global climate models.

Simple Analytic Solutions for a Convectively Driven Walker Circulation and Their Relevance to Observations

We present a linear equation for the Walker circulation streamfunction and find its analytic solutions given specified convective heating. In a linear Boussinesq fluid with Rayleigh damping and Newtonian cooling, the streamfunction obeys a Poisson’s equation, forced by gradients in the meridionally averaged diabatic heating and Coriolis force. For an idealized convective heating distribution, analytic solutions for the streamfunction can be found through an analogy with electrostatics. We use these solutions to study the response of the Walker circulation strength (mass transport) to changes in the vertical and zonal scales of convective heating. Robust responses are obtained that depend on how the total convective heating of the atmosphere responds to changing scale. If the total heating remains unchanged, increasing the zonal scale or the vertical scale always leads to a weaker circulation. Conversely, if the total heating grows in proportion to the spatial scale, the circulation becomes stronger with increasing scale. These conclusions are shown to be consistent with a three-dimensional numerical model. Moreover, they are useful in describing the observed seasonal and interannual (ENSO) variability of the Indo-Pacific Walker circulation. On both time scales, the overturning becomes weaker with increasing zonal scale of the convective region, reminiscent of our solutions where we do not vary the total convective heating. Reanalysis data also indicate that the zonal circulation is quite strongly damped, thus yielding a result that the circulation strength is directly proportional to the warm-pool spatial-mean precipitation.

Influence of Property Variation on Natural Convection in an Isothermal Vertical Finned Channel: An Extended Study

Boussineśq and non-Boussineśq fluid with thermo-physical property variation have been investigated for a vertical fin array. Computations are executed for the range of parameter such as Grashof number 1.86 × 105 and 4.42 × 105, non-dimensional S* = 0.2, 0.3, and 0.5, and non-dimensional clearance C* = 0.10, 0.15, and 0.40. The axial development of various fluid flow quantities, such as the pressure defect (P*), local Nusselt number (Nul), and the bulk temperature of the fluid (θb) has been presented for each of the property combinations. Nul is observed to have reduced by 70% near the exit from the Boussineśq fluid with fixed viscosity and thermal conductivity to the non-Boussineśq fluid with the variable property. Furthermore, the overall Nusselt number (Nu) at S* = 0.2 in an isothermal vertical fin array for an increase in Gr from 1.86 × 105 to 4.42 × 105 with different combination of properties such as Boussineśq fluid with fixed viscosity and thermal conductivity (ρb μc kc), Boussineśq fluid with variable viscosity and thermal conductivity (ρb μv kv), non-Boussineśq fluid with fixed viscosity and thermal conductivity (ρv μc kc), and non-Boussineśq fluid with variable viscosity and thermal conductivity (ρv μv kv) are observed to have an increase of 138%, 148%, 150%, and 160%, respectively.

Planetary geostrophic Boussinesq dynamics: barotropic flow, baroclinic instability and forced stationary waves

<p>Motions on planetary spatial scales in the atmosphere are governed by<br>the planetary geostrophic equations. However, not much attention has<br>been paid to the interaction between the baroclinic and barotropic<br>flow within the planetary geostrophic scaling. This is the focus of<br>the present study by utilizing planetary geostrophic equations for a<br>Boussinesq fluid supplemented by an asymptotically derived evolution<br>equation for the barotropic flow. The latter is effected by meridional<br>momentum flux due to baroclinic flow and drag by the surface wind. The<br>barotropic wind on the other hand affects the baroclinic flow through<br>buoyancy advection. By relaxing towards a prescribed buoyancy profile<br>the model produces realistic major features of the zonally symmetric<br>wind and temperature fields. We show that there is considerable<br>cancelation between the barotropic and the baroclinic surface zonal<br>mean zonal wind. The linear and nonlinear model response to steady<br>diabatic zonally asymmetric forcing is investigated. The arising<br>stationary waves are interpreted in terms of analytical solutions. We<br>also study the problem of baroclinic instability on the sphere within<br>the present model.</p><p>Reference: Dolaptchiev, S. I., Achatz, U. and Th. Reitz, 2019: Planetary<br>geostrophic Boussinesq dynamics: barotropic flow, baroclinic<br>instability and forced stationary waves, Quart. J. Roy. Met. Soc., 145: 3751-3765.</p>

Fully 3D Rayleigh-Taylor instability in a Boussinesq fluid

Rayleigh–Taylor instability occurs when a heavier fluid overlies a lighter fluid, and the two seek to exchange positions under the effect of gravity. We present a linearized theory for arbitrary three-dimensional (3D) initial disturbances that grow in time, and calculate the evolution of the interface for early times. A new spectral method is introduced for the fully 3D nonlinear problem in a Boussinesq fluid, where the interface between the light and heavy fluids is approximated with a smooth but rapid density change in the fluid. The results of large-scale numerical calculation are presented in fully 3D geometry, and compared and contrasted with the early-time linearized theory. doi:10.1017/S1446181119000087

FULLY 3D RAYLEIGH–TAYLOR INSTABILITY IN A BOUSSINESQ FLUID

Rayleigh–Taylor instability occurs when a heavier fluid overlies a lighter fluid, and the two seek to exchange positions under the effect of gravity. We present a linearized theory for arbitrary three-dimensional (3D) initial disturbances that grow in time, and calculate the evolution of the interface for early times. A new spectral method is introduced for the fully 3D nonlinear problem in a Boussinesq fluid, where the interface between the light and heavy fluids is approximated with a smooth but rapid density change in the fluid. The results of large-scale numerical calculation are presented in fully 3D geometry, and compared and contrasted with the early-time linearized theory.

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.

Boundary streaming by internal waves

Damped internal wave beams in stratified fluids have long been known to generate strong mean flows through a mechanism analogous to acoustic streaming. While the role of viscous boundary layers in acoustic streaming has been thoroughly addressed, it remains largely unexplored in the case of internal waves. Here we compute the mean flow generated close to an undulating wall that emits internal waves in a viscous, linearly stratified two-dimensional Boussinesq fluid. Using a quasi-linear approach, we demonstrate that the form of the boundary conditions dramatically impacts the generated boundary streaming. In the no-slip scenario, the early-time Reynolds stress divergence within the viscous boundary layer is much stronger than within the bulk while also driving flow in the opposite direction. Whatever the boundary condition, boundary streaming is however dominated by bulk streaming at larger time. Using a Wentzel–Kramers–Brillouin approach, we investigate the consequences of adding boundary streaming effects to an idealised model of wave–mean flow interactions known to reproduce the salient features of the quasi-biennial oscillation. The presence of wave boundary layers has a quantitative impact on the flow reversals.