The effects of alongshore variation in bottom topography on a boundary current—(topographically induced upwelling)

A theory which describes the constant f-plane flow of a steady inviscid baroclinic boundary current over a continental margin with a bathymetry that varies slowly in the alongshore but rapidly in the offshore directions is developed in the parameter regime (L_D/L)² ≤ Ro 1, where L_D is the internal deformation radius, L the horizontal length scale, and Ro the Rossby number. To lowest order in the Rossby number the flow is along isobaths with speed q_o = V_u(h,z)|Vh|/α, where V_u(h,z) is the upstream speed, α the upstream bottom slope at depth h, and Vh the bottom slope downstream at depth h. The lowest order flow produces a variation in the vertical component of relative vorticity along the isobath as the magnitude and direction of Vh vary in the downstream direction. The variation of vorticity requires a vertical as well as a cross-isobath flow at first order in the Rossby number. The first order vertical velocity is computed from the vorticity equation in terms of upstream conditions and downstream variations of the bathymetry. The density, pressure, and cross-isobath flow at first order in the Rossby number are then calculated. It is shown that in the cyclonic region of current (d/dh(V_u/α) > 0), if the isobaths diverge in the downstream direction ((∂/∂s)|Vh| < 0), then upwelling and onshore flow occur. The theory is applied to the northeastern Florida shelf to explain bottom temperature observations.     

Forces on spheres moving horizontally in a rotating stratified fluid

Abstract

Measurements have been made of the net horizontal force F acting on a sphere moving with horizontal velocity U (Reynolds numbers in the range 10²-10⁴) through a stratified fluid rotating about a vertical axis with uniform angular velocity Ω. In both homogeneous and stratified rotating fluids with small Rossby number R(R = U/Ωa ? 1 where a is the radius of the sphere) the force F is of magnitude 2ΩρUV (where ρ is the density of the fluid and V is the volume of the sphere). In a homogeneous fluid the relative directions of F and U were found to depend on the quantity F = 8Ωa ²/UD (where D is the depth of the fluid in which the object is placed (Mason, 1975)). In a rotating stratified fluid the relative directions of F and U are found to depend on the inverse Froude number k(k = Na/U where N ² = (g/δ)?ρ/?z) provided D > 4aΩ/N. In a homogeneous fluid with F ? 1 the force F is mainly in the U direction (a drag force due to inertial wave radiation) and is ～ ?0.4 |MX 2ΩρUV For F ? 1 a “Taylor column” occurs and the force, in correspondence with theoretical expectations, is ～ - 2Ω |MX UρV In a rotating stratified fluid with N ～2Ω and k ? 1 the force F is mainly in the U direction but is roughly one half of that occurring in the homogeneous situation with F ? 1 (tentatively explained as due to the evanescence of inertia-gravity disturbances). In a rotating stratified fluid with k ? 1 the flow should have no vertical motion (as with F ? 1) and again in correspondence with theoretical expectations the drag is ～ ?2 Ω |MX UρV. In a non-rotating stratified fluid the drag coefficient C _D(C _D = F U/½?ρU ²) was measured in the range k = 0.1 to 10 and had a maximum value ～ 1.2 for k ～ 3.     

Generation of superDreicer electric fields in the solar chromosphere

The electric field generation at the front of the current pulse, which originates in a coronal magnetic loop owing to the development of the Rayleigh–Taylor magnetic instability at loop footpoints, has been considered. During the τ_A ≈ l/V _A ≈ 5?25 s time (where l is the plasma plume height entering a magnetic loop as a result of the Rayleigh–Taylor instability), a disturbance related to the magnetic field tension B _?(r,t), “escapes” the instability region with the Alfvén velocity in this case. As a result, an electric current pulse I_z(z ? V _A t), at the front of which an induction magnetic field E _z, which is directed along the magnetic tube axis and can therefore accelerate particles, starts propagating along a magnetic loop with a characteristic scale of Δξ ≈ l. In the case of sufficiently large currents, when B _? ²/8π > p, an electric current pulse propagates nonlinearly, and a relatively large longitudinal electric field originates E _z ≈ 2I _z ³ V _A/c ⁴a²B_z ²l, which can be larger than the Dreicer field, depending on the electric current value.     

A steady state of the disc dynamo

Abstract

We discuss the steady states of the αω-dynamo in a thin disc which arise due to α-quenching. Two asymptotic regimes are considered, one for the dynamo numberD near the generation thresholdD ₀, and the other for |D| ? 1. Asymptotic solutions for |D—D ₀| ? |D ₀| have a rather universal character provided only that the bifurcation is supercritical. For |D| ? 1 the asymptotic solution crucially depends on whether or not the mean helicity α, as a function ofB, has a positive root (hereB is the mean magnetic field). When such a root exists, the field value in the major portion of the disc is O(l), while near the disc surface thin boundary layers appear where the field rapidly decreases to zero (if the disc is surrounded by vacuum). Otherwise, when α = O(|B|^?s) for |B| → ∞, we demonstrate that |B| = O(|D|^1/s ) and the solution is free of boundary layers. The results obtained here admit direct comparison with observations of magnetic fields in spiral galaxies, so that an appropriate model of nonlinear galactic dynamos hopefully could be specified.     

Tangential discontinuities and the optical analogy for stationary fields. v. formal integration of the force-free field equations

Abstract

This paper demonstrates the appearance of tangential discontinuities in deformed force-free fields by direct integration of the field equation ? x B = αB. To keep the mathematics tractable the initial field is chosen to be a layer of linear force-free field B_x = + B ₀cosqz, B_y = — B ₀sinqz, B_z = 0, anchored at the distant cylindrical surface ? = (x ² + y ²)^1/2 = R and deformed by application of a local pressure maximum of scale l centered on the origin x = y = 0. In the limit of large R/l the deformed field remains linear, with α = q[1 + O(l ²/R ²)]. The field equations can be integrated over ? = R showing a discontinuity extending along the lines of force crossing the pessure maximum. On the other hand, examination of the continuous solutions to the field equations shows that specification of the normal component on the enclosing boundary ? = R completely determines the connectivity throughout the region, in a form unlike the straight across connections of the initial field. The field can escape this restriction only by developing internal discontinuities.

Casting the field equation in a form that the connectivity can be specified explicitly, reduces the field equation to the eikonal equation, describing the optical analogy, treated in papers II and III of this series. This demonstrates the ubiquitous nature of the tangential discontinuity in a force-free field subject to any local deformation.     

On the spin-up of a stably stratified,electrically conducting fluid. Part 2: Spin-up controlled by the mac layer

Abstract

The linear spin-up of a stably stratified, electrically conducting fluid within an electrically insulating cylindrical container in the presence of an applied axial magnetic field is analyzed for those cases in which electric currents generated within the steady MAC layer control the fluid interior, The MAC layer is a new boundary layer first studied by Loper (1976a) which controls the fluid in the parameter range E/α² ? σS ? α²/E, α² ? 1 Where E = vωL², 2α² = σB²/pω and σS = vN²/κω;². The problem is solved using the Laplace transform and four new spin-up times are obtained. Combined into one expression they are t = ω;^?1E-½[1+(σSE/α⁶)^½ + δα^-2] [1+(σSE/α^{6 1/4}]^?1 where δ = σμv. The internal spin-up mechanisms for this problem are shown to be very similar to those discussed in part 1 (Loper, 1976b). The ten known spin-up times are summarized and their inter-relationships are investigated. It is shown how to obtain the seven hydromagnetic spin-up times from a simple torsional Alfvén wave model involving a single parameter which measures the strength of the boundary layer dissipation. Finally, the present theory is applied to the solar spin-down problem and it is found that if the magnetic field in the solar interior is at least as strong as the interplanetary field of 10^-5 gauss, then the hydromagnetic spin-down time is much shorter than the Eddington-Sweet time and is comparable to the age of the sun.     

General relationships among sound speeds: II. Theory and discussion

The dependence of bulk sound speed V_φ upon mean atomic weight $\text{m}$ and density ρ can be expressed in a single equation:

$\text{V}{}_{\mathrm{\phi }}\text{=Bρ}{}^{\mathrm{\lambda }}\text{(}\text{m}{}_0\text{m}{}^{\mathrm{<mtext>[</mtext><mtext>1</mtext><mtext>2</mtext><mtext>+\lambda (1?c)]</mtext>}}\text{(}\text{km/sec}\text{)}$

Here B is an empirically determined “universal” parameter equal to 1.42, $\text{m}{}_0\text{= 20.2}$, a reference mean atomic weight for which well-determined elastic properties exist, and λ = 1.25 is a semi empirical parameter equal to $\text{γ ?}\text{1}\text{3}$ where γ is a Grüneisen parameter. The constant c = (? ln V_M/? ln $\text{m}\text{)}{}_{\mathrm{X}}$, where V_M is molar volume, is in general different for different crystal structure series and different cation substitutions. However, it is possible to use c_Fe = 0.14 for Fe²⁺Mg²⁺ and GeSi substitutions and c_Ca ? 1.3 for CaMg substitutional series. With these values it is pos to deduce from the above equation Birch's law, its modifications introduced by Simmons to account for Ca-bearing minerals, variations in the seismic equation of state observed by D.L. Anderson, and the apparent proportionality of bulk modulus K to V_M^?4.     

Hydromagnetic waves in a differentially rotating annulus IV. Insulating boundaries

Abstract

The first three papers in this series (Fearn, 1983b, 1984, 1985) have investigated the stability of a strong toroidal magnetic field B_o =B_o(s?)Φ [where (s?. Φ, z?) are cylindrical polars] in a rapidly rotating system. The application is to the cores of the Earth and the planets but a simpler cylindrical geometry was chosen to permit a detailed study of the instabilities present. A further simplification was the use of electrically perfectly conducting boundary conditions. Here, we replace these with the boundary conditions appropriate to an insulating container. As expected, we find the same instabilities as for a perfectly conducting container, with quantitative changes in the critical parameters but no qualitative differences except for some interesting mixing between the ideal (“field gradient”) and resistive modes for azimuthal wavenumber m=1. In addition to these modes, we have also found the “exceptional” slow mode of Roberts and Loper (1979) and we investigate the conditions required for its instability for a variety of fields B_o(s?) Roberts and Loper's analysis was restricted to the case B_o∝s? and they found instability only for m=1 and ?1 <ω<0 [where ω is the frequency non-dimensionalised on the slow timescale τ_x, see (1.5)]. For other fields we found the necessary conditions to be less “exceptional”. One surprising feature of this instability is the importance of inertia for its existence. We show that viscosity is an alternative destabilising agent. The standard (magnetostrophic) approximation of neglecting inertial (and viscous) terms in the equation of motion has the effect of filtering out this instability. The field strength required for this “exceptional” mode to become unstable is found to be very much larger than that thought to be present in the Earth's core, so we conclude that this mode is unlikely to play an important role in the dynamics of the core.     

Self-consistent model of mirror structures

The purpose of this work is to give a self-consistent model of the magnetic mirrors using a perturbative magnetohydrostatic approach. With the help of this model a number of features have been revealed like geometry, stability and behavior for different temperature anisotropies (A=T_⊥/T_||). The basic relations we use in order to derive the model for the mirror structures are the magnetohydrostatic equilibrium condition and an expression for the anisotropy in the case of bi-Maxwellian distribution (Lee et al., J. Geophys. Res. 92 (1987) 2343). Based on these equations, we have found analytical expressions for the magnetic field (δB), pressure (δp) and temperature (δT) perturbations. From the investigation of the dependence of the magnetic mirrors on the unperturbed anisotropy (A₀), we have found the well-known behavior (opposite phase variations of the magnetic field intensity and number density) for A₀>1 (Tsurutani et al., Geophys. Res. 87 (1982) 6060). For A₀<1, the behavior is different but the mirror structures still exist. However, if the anisotropy is in a range of values depending on the plasma parameter β_0⊥=p_0⊥/(B₀²/2μ₀), the magnetic mirrors can no longer exist. From the comparison between the current density deduced from the Ampere law, necessary to sustain the magnetic mirror, and the gradient-curvature drift current density actually being inside the magnetic mirror, we have been able to determine instability regions in the (A₀,β_0⊥)-plane.     

On the spin-up of a stably stratified,electrically conducting fluid. Part 1: Spin-up controlled by the hartmann layer

Abstract

The linear spin-up of a stably stratified, electrically conducting fluid within an electrically insulating cylindrical container in the presence of an applied axial magnetic field is analyzed for those cases in which electric currents generated within the steady Hartmann boundary layer control the fluid interior. It is shown how to obtain the known spin-up times for a homogeneous, nonconducting fluid (τ = E ^-½), a stably stratified, nonconducting fluid (τ = (σS/E, E ^?1) and a homogeneous conducting fluid (τ = α^?1 E ^-½) from the present formulation where τ = v/ωt, E = v/ωL ², σS = vN²/κω² and 2α² = σB²/pω. The problem is solved in the parameter range E?α²?1, α²/E?σS using the Laplace transform and two new spin-up times are obtained. Combined into one expression, they are τ = (1 + δ)α^?1E^-½ where δ = σμv. The spin-up mechanism is investigated and it is found that, in contrast to the homogeneous, conducting case, torsional Alfvén waves may be instrumental in the spin-up of a stratified conducting fluid. The effects of viscous and ohmic diffusion on the torsional Alfvén wave fronts are studied and the following regimes are identified: 0 < δ ?E/α², spin-up by meridional circulation of electric current with no Alfvén waves; E^-½/α ? δ ? 1, spin-up by meridional circulation of electric current with transient Alfvén waves; α/E^½ ? δ ? α²/E, spin-up by meridional circulation of current with weak Alfvén waves; 1 ? δ ? α/E^½, spin-up by strong Alfvén waves; α^½/E ? δ ? α²/E, spin-up by viscous diffusion with transient Alfvén waves; α/E ? δ < ∞, spin-up by viscous diffusion with no Alfvén waves.     

Two-dimensional density currents in a confined basin

We present new experimental results on the mechanisms through which steady two-dimensional density currents lead to the formation of a stratification in a closed basin. A motivation for this work is to test the underlying assumptions in a diffusive “filling box” model that describes the oceanic thermohaline circulation (Hughes, G.O. and Griffiths, R.W., A simple convective model of the global overturning circulation, including effects of entrainment into sinking regions, Ocean Modeling, 2005, submitted.). In particular, they hypothesized that a non-uniform upwelling velocity is due to weak along-slope entrainment in density currents associated with a large horizontal entrainment ratio of E _eq ?～?0.1. We experimentally measure the relationship between the along-slope entrainment ratio, E, of a density current to the horizontal entrainment ratio, E _eq, of an equivalent vertical plume. The along-slope entrainment ratios show the same quantitative decrease with slope as observed by Ellison and Turner (, 6, 423–448.), whereas the horizontal entrainment ratio E _eq appears to asymptote to a value of E eq?=?0.08 at low slopes. Using the measured values of E _eq we show that two-dimensional density currents drive circulations that are in good agreement with the two-dimensional filling box model of Baines and Turner (Baines, W.D. and Turner, J.S., Turbulent buoyant convection from a source in a confined region, J. Fluid. Mech., 1969, 37, 51–80.). We find that the vertical velocities of density fronts collapse onto their theoretical prediction that U =-2^?2/3 B ^1/3 E _eq ^2/3 (H/R) ζ, where U is the velocity, H the depth, B the buoyancy flux, R the basin width, E _eq the horizontal entrainment ratio and?ζ?= z/H the dimensionless depth. The density profiles are well fitted with?Δ?= 2^?1/3 B ^2/3 E _eq ^?2/3 H _-1 [ln(ζ )?+?τ ], where?τ?is the dimensionless time. Finally, we provide a simple example of a diffusive filling box model, where we show how the density stratification of the deep Caribbean waters (below 1850?m depth) can be described by a balance between a steady two-dimensional entraining density current and vertical diffusion in a triangular basin.     

MAGNETOTELLURIC RESPONSE ON VERTICALLY INHOMOGENEOUS EARTH HAVING CONDUCTIVITY VARYING LINEARLY WITH DEPTH*

Magnetotelluric response is studied for an inhomogeneous medium having conductivity varying linearly with depth as σ(z) =σ₁+αz. For a medium having conductivity increasing linearly with depth, the phase of the impedance approaches 60° at long periods and the apparent resistivity becomes log (ρ_a) = 2 log (1.36/α^1/3) — 1/3 log (T'). The asymptote of log (ρ_a, T'→∞) when plotted against log (T') has a constant gradient —1/3 and has an intercept on the log (T') axis, which equals 6 log (1.36/α^1/3). When a homogeneous layer with a moderate thickness overlies an inhomogeneous half-space, this layer does not affect the asymptote, but it affects the cut-off period and pushes this toward the long period direction. For a medium having conductivity decreasing linearly with depth, the impedance is equivalent to that of a Cagniard two-layer model; the intercept period related to the thickness is T'₀=σ₁(h₂/2)². Homogeneous multilayer approximations to an inhomogeneous layer are also investigated, and it is shown that the fit to the model variation depends on the number of layers and the layer parameters chosen.     

A laboratory study of baroclinic instability

Abstract

Experimental and theoretical results are presented for a simple system which exhibits baroclinic instability. We consider the motion of two immiscible fluids with densities ρ ₁ and ρ ₂ contained in a cylinder rotating with angular frequency ω. The motion is driven by a contact lid rotating with frequency ω + ω. In this paper ω, ω, 2(ρ ₂ – ρ ₁)/(ρ ₂ + ρ ₁), and the geometry are such that the interface does not intersect the “ground” (e.g. an almost horizontal boundary). The motions are described by two-layer quasi-geostrophic equations which are identical, except perhaps for the presence of interfacial friction and tension, with those used in meteorology and oceanography. For small enough internal Froude number F = 4ω² L ²/(g(Δρ/ρ)H) or small enough Rossby number ? = ω/2ω the flow is steady and axisymmetric, the velocity field in each layer being determined primarily by frictional effects in top, bottom, and interfacial Ekman layers. For certain (F, ?) the flow becomes non-axisymmetric. The transition points for the case where the basic potential vorticity gradient is due to interface slope alone have been carefully measured and are in very good agreement with a linear instability theory which neglects sidewall effects. Some preliminary observations of supercritical motion, which include repeatable amplitude and wavenumber vacillation, are reported.     

Two-and three-dimensional linear convection in a rotating annulus

Abstract

An exceptional case to the model-independent theory of Knobloch (1995) is presented, by investigating a rotating cylindrical annulus of height H and side wall radii r _o and r _i, with non-slip, perfectly thermally conducting side walls and thermally insulating stress-free ends. Radial heating permits the possibility of either two- or three-dimensional convective solutions being the preferred mode. An analytical solution is obtained for the two-dimensional case and a numerical solution for the three-dimensional solution, which is also applied to the two-dimensional solution. It is shown that both two- and three-dimensional solutions can be realized depending on the aspect ratio, γ = H/d, where d = r _o-r _i is the thickness of the annulus, the radii ratio λ = r _i/r _o and the rotation rate of the model. For γ = O(1) and λ = 0.4, the preferred convective solution is three-dimensional when the Taylor number, T < 10² and two-dimensional for T > 10². For small aspect ratios, γ ? 1, the preferred mode is two-dimensional for all rotation rates.     

The maximum-slope method for the interpretation of electrical soundings over two horizontal layers

Using the Carpenter configuration, in which the distances between the successive electrodes of a four-electrode system are kept at r, s and r (as shown in Fig. 1), where r = ma and s = la, or the Wenner-Carpenter configuration, in which r = s, the corresponding formulae for the apparent resistivities ρ_A^α, ρ_A^β and ρ_A^γ for the α, β and γ configurations are used to compute maximum gradient values for various values of the reflection coefficient K. For any observed curve for these resistivity types, the maximum slope is determined. A set of standard curves is presented which, provided that K is known, then allows the thickness of the top layer to be determined.In the case that K is unknown, another set of standard curves is presented allowing determination of K provided that the maximum slopes of any two of the three resistivity curves are known.The method has been tested for both field and laboratory observations. Discrepancies between the observed and actual values of resistivity are noted and attributed partly to instrumental errors.     

A comparison of numerical simulations of eddy generation performed with a two- and a three-layer quasi-geostrophic model of oceanic mesoscale eddies

Abstract

The generation of eddies by a large-scale flow over mesoscale topography is studied with the help of two- and three-layer nonlinear quasi-geostrophic models of the open ocean. The equations are integrated forward in time with no eddies present initially. For a given time, the displacement of the interface between layers two and three (ζ) tends to a well-defined limit (function of the horizontal spatial coordinates) as ρ ₃ - ρ ₂ → 0 (ρ_r is the density of layer r). Even for values of α[= (ρ ₃ - ρ ₂)/(ρ ₂ - ρ ₁)] ^as small as 0.01 the potential energy due to ζ is not negligible and it can reach, in some cases, a considerable fraction of the total eddy energy.     

Radiation damping in the context of one-dimensional wave propagation: A teaching perspective

This paper has two objectives: to connect directly radiation damping and 1D elastic wave propagation, and to create a simple teaching tool to introduce the subject to students and engineers trained in Structural Dynamics. The first objective is achieved by obtaining the equivalent radiation modal damping using time domain solutions for the fundamental mode in shear of an elastic layer (soil) on flexible rock, for the case in which the rock–soil Impedance Ratio in shear, I=(ρ_rV_r)/(ρ_sV_s)>1, where ρ=mass density and V=shear wave velocity. These time domain solutions are developed for the case of steady-state input sinusoidal shear waves propagating vertically in the rock as well as for horizontal free vibration of the layer. Both derivations result in the same approximate expression for the modal radiation damping in the first mode, ζ_r1≈2/(πI), which is in turn identical to the approximate equation obtained by Roesset and Whitman [11] using a frequency domain approach. This expression for ζ_r1 is linked to the fact that, during free vibration, the ratio between two successive positive displacement peaks u_so and u_s1 at the ground surface is u_s0/u_s1=[(1+I)/(1−I)]², associated with two wave reflections at the soil–rock interface. From this ratio, and after applying the standard expression to obtain modal damping from damped free vibration, the same expression for ζ_r1 is reached again, ζ_r1≈[1/(2π)] ln(u_s0/u_s1)≈[1/(2π)] ln[(1+I)/(1−I)]²≈2/(πI). This finding allows development of the simple teaching tool proposed at the end of the paper. While only a crude approximation lacking in rigor, this teaching tool is physically intuitive, links directly wave propagation and modal damping in a simple way and gives the correct result.     

DIELECTRIC PROPERTIES OF WET SOILS IN THE FREQUENCY RANGE 1–3000 MHz1

The effective relative dielectric constant ?_{e, r} and the effective conductivity σ_e have each been determined as a function of frequency in the range 1–3000 MHz at volumetric water contents of up to approximately 0.74 for clays, 0.83 for a peat and 0.56 for a silt. At frequencies above about 25 MHz (depending on soil type), ?_{e, r}increases with water content for all samples. However, at lower frequencies, ?_{e, r}only increases with water content as long as the wet density also increases, which is the case for water contents up to a critical value lying between 0.35 and 0.48. At higher water contents, ?_{e, r}and the wet density decrease with increasing water content. Consequently, curves of ?_{e, r}versus frequency for two wet samples with different water contents, at least one of them higher than the critical value, are seen to cross at about 25 MHz. Below the critical value the curve of the sample with the lower water content is below the other curve at all freqencies applied. At a given frequency, σ_e has a maximum as a function of water content. This is tentatively explained by assuming that σ_e is the sum of pore water conductivity (increasing with water content until all salts in the soil are dissolved into the water and then decreasing) and surface water conductivity (increasing with wet density and therefore increasing with water content up to the critical value and then decreasing). At frequencies higher than 1000 MHz, ?_{e, r}depends only weakly on salinity (which is represented by the measured conductivity). It shows an increasing dependence if the frequency is decreased towards 1 MHz. The highest values of ?_{e, r}and σ_e, measured in this work, occur for a sample of wet, nearly saturated silt originating from a location below sea-level near to the Dead Sea, Israel: ?_{e, r}decreases continuously from a value of about 10⁴ at 3 MHz to about 10² at 200 MHz, while σ_e rises from about 4 S/m to 5 S/m at these respective frequencies. The dependence of the wavelength on the loss-tangent is strong and the wavelength is considerably smaller than it would be in a dielectric. This is the only sample for which σ_e increases with water content, even if the latter is above its critical value. Therefore it is assumed that the pore water conductivity is greater than the surface water conductivity if the volumetric water content is lower than 0.564, the maximum value applied. The measurements give evidence for the presence of a relaxation at about 3 MHz for all samples examined.