Space-Time scales of internal waves

Abstract

We have contrived a model E(αω) α μ^?1ω^?p+1(ω ²?ω _i ²)^?+ for the distribution of internal wave energy in horizontal wavenumber, frequency-space, with wavenumber α extending to some upper limit μ(ω) α ω ^r-1 (ω ²?ω _i ²)^½, and frequency ω extending from the inertial frequency ω _i to the local Väisälä frequency n(y). The spectrum is portrayed as an equivalent continuum to which the modal structure (if it exists) is not vital. We assume horizontal isotropy, E(α, ω) = 2παE(α₁, α₂, ω), with α₁, α₂ designating components of α. Certain moments of E(α₁, α₂, ω) can be derived from observations. (i) Moored (or freely floating) devices measuring horizontal current u(t), vertical displacement η(t),…, yield the frequency spectra F _(u,η,…)(ω) = ∫∫ (U ², Z ²,…)E(α₁, ∞₂, ω) dα₁ dα₂, where U, Z,… are the appropriate wave functions. (ii) Similarly towed measurements give the wavenumber spectrum F _(…)(α1) = ∫∫… dα₂ dω. (iii) Moored measurements horizontally separated by X yield the coherence spectrum R(X, ω) which is related to the horizontal cosine transform ∫∫ E(α1, α2 ω) cos α₁ Xdα₁ dα₁. (iv) Moored measurements vertically separated by Y yield R(Y, ω) and (v) towed measurements vertically separated yield R(Y, α₁), and these are related to similar vertical Fourier transforms. Away from inertial frequencies, our model E(α, ω) α ω ^?p-r for α ≦ μ ω ω ^r, yields F(ω) ∞ ω ^?p, F(α₁) ∞ α₁ ^?q, with q = (p + r ? 1)/r. The observed moored and towed spectra suggest p and q between 5/3 and 2, yielding r between 2/3 and 3/2, inconsistent with a value of r = 2 derived from Webster's measurements of moored vertical coherence. We ascribe Webster's result to the oceanic fine-structure. Our choice (p, q, r) = (2, 2, 1) is then not inconsistent with existing evidence. The spectrum is E(∞, ω) ∞ ω ^?1(ω ²?ω _i ^{2 ?1}, and the α-bandwith μ ∞ (ω ²?ω _i ²)⁺ is equivalent to about 20 modes. Finally, we consider the frequency-of-encounter spectra F([sgrave]) at any towing speed S, approaching F(ω) as S ≦ S _o, and F(α₁) for α₁ = [sgrave]/S as S ≧ S _o, where S _o = 0(1 km/h) is the relevant Doppler velocity scale.     

Vertical mixing in Überlinger See,western part of Lake Constance

Depth variable vertical eddy diffusion coefficients for heat (K _z) were calculated from continuously measured temperature profiles in Überlinger See (western part of Lake Constance). The temperatures were averaged over vertical intervals of 10 m yielding 14 discrete values (maximum depth of Überlinger See: 147 m). A linear fit from 10 June to 29 September 1987 was used to smooth the significant temperature fluctuations caused by internal seiches of Lake Constance.Assuming horizontal homogeneity for the smoothed data the Gradient-Flux-Method was applied to compute vertical diffusion coefficientsK _z at different depths using the depth variable volumes and surfaces of the 14 layers. The resulting mean diffusion coefficients for the period from June to September are 0.04 cm²/s near the thermocline and up to 0.8 cm²/s in deeper strata (accuracy: ± 50%). It is shown that horizontal mixing between Überlinger See and Obersee (main lake) alters the computation ofK _z by less than 50%.A relationship betweenK _z and stability (Brunt-Väisälä) frequencyN is found which corresponds well to the theory of internal wave induced turbulence.Combining the diffusion coefficients with measured phosphorus profiles, a phosphorus flux from the hypolimnion to the epilimnion of (0.7 ± 0.4) mg P m^–2 d^–1 was calculated, corresponding to about 20% of the average external loading per area of Lake Constance in 1986.     

APPROXIMATE INVERSION OF AIRBORNE EM DATA FROM A MULTILAYERED GROUND1

A complex transfer function c (or generalized skin depth) can be derived from data for the secondary magnetic field measured by a dipole system with small coil spacing at height h above the ground. This function has a useful property: For a uniform or layered ground, the real part of c yields the‘ centroid depth’z* of the in-phase current system as a function of frequency. This parameter can be combined with the apparent resistivity ρ_a derived by conventional methods. The function ρ_a(z*), if known over a broad frequency range, yields a smoothed approximation of the true distribution ρ(z) without an initial model. The relations between ρ_a(z*) and ρ(z) are studied for a number of multilayer models. An example of the application of the ρ_a*) algorithm to data from a groundwater survey is given.     

Analytical solution of transient flow in a sloping soil layer with recharge

Abstract

An analytical solution of planar flow in a sloping soil layer described by the linearized extended Boussinesq equation is presented. The solution consists of the sum of steady-state and transient-series solutions, the latter in a separation-of-variables form, and can satisfy an arbitrary initial condition via collocation; this feature reduces the number of series terms, making the solution efficient. Key parameter is the dimensionless linearization depth η _o (R), R being the dimensionless recharge. The variable η _o (R), not the slope, characterizes the flow as kinematic or diffusive, and R ≈ 0.2 demarcates the two regimes. The transient series converges rapidly for large η _o (large R, near-diffusive flow) and slowly as η _o → 0 (kinematic flow). The quasi-steady (QS) state method of Verhoest & Troch is also analysed and it is shown that the QS depth profiles approximate the transient ones well, only if Δt exceeds a system-dependent transition time between flow states (possibly >>1 day). In an application example for a 30-day recharge series, the QS solution differs from the transient one by as much as 20% (RMSE = 15%), does not track recharge changes as well and fails to conserve mass.     

Estimation of sedimentary layer shear wave velocity using micro-tremor H/V ratio measurements for Bangalore city

For assessing earthquake hazard of metro cities, knowledge of soil amplification, thickness and properties of sedimentary layer are essential. In order to map the soil thickness using microtremor survey method, in Bangalore city, it is required to calibrate the relation between fundamental resonance frequency of the soil layer and its thickness for the region. For this purpose microtremor survey was carried out at 34 locations in the city where borehole log was available. The resonance frequency of the soil is evaluated from the microtremor recordings using the H/V ratio technique. A nonlinear regression relation between the thickness of sedimentary layer h (m), from the borehole logs, and the resonance frequency f_r (Hz), was derived as h=(58.3±8.8)f_r^{−(0.95)±0.1}. Using the model of shear wave velocity increasing with depth at these locations, the derived average shear wave velocity and the corresponding soil thickness were used, to get an empirical relation between V_S (m/s) and depth z(m), as V_s=(174±28)(1+z)^0.16±0.07. This relation also compares reasonably with the fit obtained between simulated V_S and depth from borehole logs for Bangalore city. The calibrated relations can be used at locations in Bangalore city where borehole logs are not available, for finding the thicknesses and shear wave velocities of the local soil layers at the survey locations.     

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.     

Modelling of magnetic data for scaling geology

Interpretation of magnetic data can be carried out either in the space or frequency domain. The interpretation in the frequency domain is computationally convenient because convolution becomes multiplication. The frequency domain approach assumes that the magnetic sources distribution has a random and uncorrelated distribution. This approach is modified to include random and fractal distribution of sources on the basis of borehole data. The physical properties of the rocks exhibit scaling behaviour which can be defined as P(k) = Ak^?β, where P(k) is the power spectrum as a function of wave number (k), and A and β are the constant and scaling exponent, respectively. A white noise distribution corresponds to β = 0. The high resolution methods of power spectral estimation e.g. maximum entropy method and multi‐taper method produce smooth spectra. Therefore, estimation of scaling exponents is more reliable. The values of β are found to be related to the lithology and heterogeneities in the crust. The modelling of magnetic data for scaling distribution of sources leads to an improved method of interpreting the magnetic data known as the scaling spectral method. The method has found applicability in estimating the basement depth, Curie depth and filtering of magnetic data.     

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.     

Finite-amplitude mountain waves in a compressible atmosphere including the effects of dissipation

Abstract

Adiabatic, two-dimensional, steady-state finite-amplitude, hydrostatic gravity waves produced by flow over a ridge are considered. Nonlinear self advection steepens the wave until the streamlines attain a vertical slope at a critical height z_c. The height z_c , where this occurs, depends on the ridge crest height and adiabatic expansion of the atmosphere. Dissipation is introduced in order to balance nonlinear self advection, and to maintain a marginal state above z_c. The approach is to assume that the wave is inviscid except in a thin layer, small compared to a vertical wavelength, where dissipation cannot be neglected. The solutions in each region are matched to obtain a continuous solution for the streamline displacement δ. Solutions are presented for different values of the nondimensional dissipation parameter β. Eddy viscosity coefficients and the thickness of the dissipative layer are expressed as functions of β, and their magnitudes are compared to other theoretical evaluations and to values inferred from radar measurements of the stratosphere.

The Fourier spectrum of the solution for z ≫ z_c is shown to decay exponentially at large vertical wave numbers n. In comparison, a spectral decay law n ^?-8/3 characterizes the marginal state of the wave at z = z_c .     

Zur Rolle der Stickstoffixation im Stoffhaushalt eines geschichteten Sees

In the dimict lake Arend (5.1 km², 146 hm³, 49.5 m z_max), nitrogen is production-limiting with concentrations below the detection limit during the production period. Phytoplankton achieves biomasses of up to 18 mg/l fresh matter, essential contributions being made by Aphanizomenon with 2 mg/l and Anabaena with up to 10 mg/l. Nitrogen fixation was measured by the ethine reduction technique (acetylene reduction) during periods of the occurrence of heterocystforming Cyanophyceae and achieved peak values up to 6.59 μg N² · h^?1 · l^?1 or 14.87 m^?2 · h^?1 g N₂ · m^?2 · h^?1. The rates of fixation show a safe correlation with the biomass of heterocyst-containing Cyanophyceae (r = 0.88), their development beginning at values below the N : P-ratio of 2.66.     

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.     

Hyporheic exchange mechanism driven by flood wave

To study the hyporheic exchange driven by a single peak flood-induced water level fluctuation (i.e., flood wave), a method combining numerical simulation with theoretical derivation was proposed based on the Inbuk Stream, Korea, where flooding occurs frequently. The hyporheic exchanges induced by different flood waves were investigated by varying amplitude (A), duration (T), wave type parameter (r), and rising duration (t_p), which were adopted from the real-time stream stage fluctuations. Additionally, the idea of constant upstream flood volume (CUFV) condition for flood waves was put forward, and the effects of “Botan” (T/A) and peak number (N) on hyporheic exchange were studied. The results showed that the hyporheic exchange flux (q) was controlled by the water level h (sine-type) and its change rate v (cosine-type), and was proportional to the polynomial of them q ∝ (ω∙ h + v), where ω is the angular frequency of the flood wave. Based on this mechanism, the influence principles on hyporheic exchanges of the typical flood wave parameters (A, T, r and t_p) as well as T/A and N under CUFV condition were clarified. The main characteristic variables of hyporheic exchange, which were maximum aquifer storage and residence time, were positively correlated. They also had positive relations to the integral of the flood wave over time, which increased when the wave became higher, wider, rounder and less skewed. However, when CUFV condition was imposed, the residence time was positively correlated with T/A, whereas the maximum aquifer storage was negatively correlated with T/A. With the increase in N, water exchanged more frequently and some water returned to the stream early, leading to the slight decrease in maximum aquifer storage and residence time. These findings enriched the theory of hyporheic exchange driven by surface water fluctuation and be of great significance to enhance pollutant degradation in the hyporheic zone downstream of reservoirs.     

New electron energy transfer rates for vibrational excitation of N2

In this paper we present the results of a study of the electron cooling rate, the production rates of vibrationally excited N₂(v), and the production frequency of the N₂ vibrational quanta arising from the collisions of electrons with unexcited N₂(0) and vibrationally excited N₂(1) molecules as functions of the electron temperature. The electron energy transfer rates for vibrational excitation of N₂ have been calculated and fit to analytical expressions by use of the revised vibrationally excited N₂ cross sections. These new analytical expressions are available to the researcher for quick reference and accurate computer modeling with a minimum of calculations.     

Letter to the editor Electric field fluctuations (25–35 min) in the midnight dip equatorial ionosphere

Measurements with a HF Doppler sounder at Kodaikanal (10.2°N, 77.5°E, geomagnetic latitude 0.8°N) showed conspicuous quasi-periodic fluctuations (period 25/35 min) in F region vertical plasma drift, V_z in the interval 0047/0210 IST on the night of 23/24 December, 1991 (A_p = 14, K_p < 4^–). The fluctuations in F region vertical drift are found to be coherent with variations in B_z (north-south) component of interplanetary magnetic field (IMF), in geomagnetic H/X components at high-mid latitude locations both in the sunlit and dark hemispheres and near the dayside dip equator, suggestive of DP2 origin. But the polarity of the electric field fluctuations at the midnight dip equator (eastward) is the same as the dayside equator inferred from magnetic variations, contrary to what is expected of equatorial DP2. The origin of the coherent occurrence of equatorial electric field fluctuations in the DP2 range of the same sign in the day and night hemispheres is unclear and merits further investigations.     

Low frequency 1/f-like fluctuations of the AE-index as a possible manifestation of self-organized criticality in the magnetosphere

Low frequency stochastic variations of the geomagnetic AE-index characterized by 1/f^b-like power spectrum (where f is a frequency) are studied. Based on the analysis of experimental data we show that the B_z-component of IMF, velocity of solar wind plasma, and the coupling function of Akasofu are insufficient factors to explain these behaviors of the AE-index together with the 1/f^b fluctuations of geomagnetic intensity. The effect of self-organized criticality (SOC) is proposed as an internal mechanism to generate 1/f^b fluctuations in the magnetosphere. It is suggested that localized spatially current instabilities, developing in the magnetospheric tail at the initial substorm phase can be considered as SOC avalanches or dynamic clusters, superposition of which leads to the 1/f^b fluctuations of macroscopic characteristics in the system. Using the sandpile model of SOC, we undertake numerical modeling of space-localized and global disturbances of magnetospheric current layer. Qualitative conformity between the disturbed dynamics of self-organized critical state of the model and the main phases of real magnetospheric substorm development is demonstrated. It is also shown that power spectrum of sandpile model fluctuations controlled by real solar wind parameters reproduces all distinctive spectral features of the AE fluctuations.     

ESTIMATION OF DEPTH TO CONDUCTORS BY THE USE OF ELECTROMAGNETIC TRANSIENTS*

The transient response of a layered structure to plane wave excitation can be considered to be composed of a series of waves and a ground wave. For the case of a half-space of conductivity σ and permeability μ the maximum in the electric field is found at a depth z and time t when t=z²σμ/2. This formula can be used to estimate the depth to a buried horizontal conductor with an accuracy that depends upon the resistive contrast at the conductor's surface. The above ray type of solution can be converted to a solution composed of a number of modes by the use of a Poisson transform and the transformed solutions yield decay constants that are consistent with the previously reported results. In the case of a finite source, the maximum in the electric field is strongly directed. The direction depends upon the geometry of the source and the air-earth interface. Although the maximum varies with direction it can be shown that in some directions similar laws to that above are valid. The depth to a conductor can be estimated from the early part of the transients when the ground wave is removed. The removal of the ground wave from the transient is facilitated by the use of an apparent conductivity formula. Although these results were obtained under restrictive conditions they do provide some insight into the electrical transients that are encountered by studying more complex models.     

Fundamental geodetic parameters of the earth's figure and the structure of the earth's gravity field derived from satellite data

Summary Using the geocentric constant GM=398 601.3 × 10 ⁹ m ³s ^–2 , the known value of the angular velocity of the Earth's rotation , Stokes' constants J _n ^(k) and S _n ^(k) upto n=21 (zonal), n=16 (tesseral and sectorial) [2], the geocentric co-ordinates and heights above sea-level of SAO satellite stations [2], the following will be derived: the potential on the geoid W_o, the scale factor for lengths R_o=GM/W_o, the radius-vector of the surface W=W_o, the parameters of the best-fitting Earth tri-axial ellipsoid, and the components of the deflections of the vertical with respect to the geocentric rotational IAG ellipsoid (Lucerne 1967), as well as to the best-fitting geocentric tri-axial ellipsoid. Some of the differences in the structure of the gravity field over the Northern and Southern Hemispheres will be given, and the mean values of gravity over the equatorial zone, determined from the dynamics of satellite orbits, on the one hand, and from terrestrial gravity data, on the other, will be compared.Presented at the Fifteenth IUGG General Assembly, Moscow, July 30 — August 14, 1971.     

THE INDUCTIVE RESPONSE OF A HORIZONTAL CONDUCTING CYLINDER BURIED IN A UNIFORM EARTH FOR A UNIFORM INDUCING FIELD*

The inductive response of a conducting horizontal cylinder embedded in a uniform earth is studied using numerical results obtained for an analytical solution for the problem of a conducting cylinder buried in a homogeneous earth for the case of a uniform inducing field. A check of the validity of the numerical results is made by a comparison with analogue model measurements for a number of cases. Numerical results for a range of cylinder radii (a = 1–10 km), depths of burial (d= 0–4 km), conductivity contrasts (σ₂= 10^?2-10 Sm^?1), and source frequencies (f= 10^?1-10^?4 Hz) of interest in the interpretation of magnetotelluric field measurements are presented. The results indicate that for a uniform inducing field the conductivity and depth of burial of a horizontal cylindrical inhomogeneity are best determined through a measurement of the amplitudes H_y, H_z and E_x and the phases φ_y and Ψ_x.