The scattering of Rossby waves from finite abrupt topography

The scattering of first mode linear baroclinic Rossby waves by a top-hat ridge in a continuously stratified ocean, with Brunt-Väisälä frequency that decays exponentially with depth below a surface mixed layer, is the subject of this study. A numerical mode matching technique is used to calculate the transmission coefficients for the propagating modes over the ridge. It is found that the scattered field depends crucially upon the stratification. For example, when the majority of the density variation is confined to a thin thermocline, corresponding to a small e-folding scale, gamma ^?1, for the Brunt-Väisälä frequency, a large amount of the incident wave energy is reflected by a small amplitude ridge. Appreciable energy conversion between the propagating barotropic and baroclinic modes takes place in this case. An asymptotic analysis for a small amplitude ridge is presented that confirms these numerical results. In the limit gamma ^?1→ 0, it is demonstrated that the scattered field in the continuously stratified ocean model differs markedly from the two-layer solution. The latter does not exhibit appreciable reflection of the incident wave energy for a small amplitude ridge. In conclusion, the application of a two-layer ocean model to describe Rossby wave scattering by ridges in place of a continuously stratified model cannot be recommended.     

Boundary layer models of ocean floor spreading

Summary The equations of conservations of momentum and energy scaled with the characteristic values of the mantle indicate the presence of the upper boundary layer to produce the estimated rate of the ocean floor spreading by convection and the importance of the frictional heating. The depth of the upper boundary layer can be estimated from the balance of the viscous force with the horizontal pressure gradient at the sea floor. It is of the orders of 100 km and becomes deeper for the Pacific than for the Atlantic Ocean and also with frictional heating than without it. The frictional heating increases the surface heat flow of the heat conduction by ten to twenty percent for the Pacific Ocean but only by a few percent for the Atlantic Ocean. The similarity solutions are determined for the temperature and horizontal velocity in the upper boundary layer. These solutions are expressed in power series of the variabley x ^–n , wherex, y, andn are horizontal and vertical coordinates and numerical constant, respectively. Both temperature and horizontal velocity within the boundary layer are higher for the Pacific than for the Atlantic Ocean. When a larger viscosity is applied, it causes the increase of horizontal velocity below the surface because of the surface boundary conditions of the finite velocity and of vanishment of the velocity shear. The higher horizontal velocity generates higher temperature because it advects hotter material from the mid-ocean ridge site. The direct effect of frictional heating on the temperature distribution of the similarity solution is almost negligible, since the shear zone is deep and near the lower boundary of the upper boundary layer. In the similarity solution, the surface heat flow which is increased by the frictional heating is given as the boundary value. The effect of the frictional heating is important below the mid-ocean ridge.     

Rotating flow over long shallow ridges

Abstract

The flow of a rotating homogeneous, incompressible fluid past a long ridge is investigated. An analysis is presented for flows in which E ? 1, Ro ～ E^½, H/D ～ E⁰, h/D ～ E^½ and cosα ～ E⁰ where E is the Ekman number, Ro the Rossby number, H/D the fluid depth to ridge width ratio, h/D the ridge height to ridge width ratio and α the angle between the free stream flow and a line perpendicular to the ridge axis. The analysis includes effects of the nonlinear inertial terms. Particular examples of a ridge of triangular cross section and a sinusoidal topography are investigated in some detail. Experiments are presented for a triangular ridge which are in good agreement with the theory.     

Linear theory of the response of a two layer ocean to a moving hurricane‡

Abstract

This paper describes the linear response of an inviscid two‐layer model of a deep ocean on an f‐plane to a hurricane translating across the surface at constant speed. The forcing is a localized, radially‐symmetric pattern of positive wind stress curl and negative pressure anomaly. Only the steady state response is considered. The principal result is the identification of an internal wake in the lee of the storm, present when the translation speed of the storm exceeds the baroclinic long wave speed. The amplitude of the wake depends on the length of time over which the stress is experienced at a given point. The angle of the wedge filled by the wake is small, an effect due to the fact that the scale of a hurricane is typically larger than the baroclinic radius of deformation. After the wake disperses, a geostrophically balanced baroclinic ridge remains along the storm track.     

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 .     

An analytical model for ice-edge upwelling

Abstract

This paper presents an analytical, two-dimensional model of the wind-induced homogeneous circulation near the edge of an ice pack floating on the ocean surface. It is shown that a vertical shear layer arises under the ice edge, by which the wind-driven geostrophic motion in the open ocean is matched to the flow region underneath the ice. As in coastal upwelling models, this shear layer consists of a thin E ^1/2-layer inside a thicker E ^1/4-layer (E being the Ekman number). Under certain conditions the shear layer produces a vertical mass flux from the bottom to the surface Ekman layer. Near the surface this upwelling flux is concentrated in the narrow E ^1/2-layer. Comparison with observations of upwelling at the edge of a polar ice pack shows good agreement.     

Penetrative convection in the upper ocean due to surface cooling

Abstract

A new non-linear model of mixing and convection based on a modelling of two buoyant interacting fluids is applied to penetrative convection in the upper ocean due to surface cooling. In view of simple algebra, the model is one-dimensional. Dissipation is included, but no mean shear is present. A non-similar analytical solution is found in the case of a well-mixed layer bounded below by a sharp thermocline treated as a boundary layer. This solution is valid if the Richardson number, R _i , defined as the ratio of the total mixed-layer buoyancy to a characteristic rms vertical velocity, is much greater than unity. The model predicts a deepening rate proportional to R _i ^?3/4. The thermocline remains of constant thickness, and the ratio thermocline thickness to mixed-layer depth decreases as R _i ^?3/4 as the mixed layer deepens. If the surface flux is constant, the mixed-layer depth increases with time as t ^½. The vertical structure throughout the mixed layer and thermocline is given by the analytical solution, and vertical profiles of mean temperature and vertical fluxes are plotted. Computed profiles and available laboratory data agree remarkably well. Moreover, the accuracy of the simple analytical results presented here is comparable to that of sophisticated turbulence numerical models.     

Research on geomagnetic tectonic characteristics of the crust in Haicheng seismic area

In this paper, the two metnods, rectangular spectrum (RS) and maximum entropy power spectrum (MES), have been used in calculating the geomagnetic structures of the crust in Haicheng and its adjacent areas and carrying out the geomagnetic stratification of the crust. The result of the research indicates that the crust can be devided into such three layers as nonmagnetic layer, magnetic layer and regressive magnetic layer from the top to the bottom. It is also found the distribution of the geomagnetic bottom interface (Curie surface) is consistent with the lower interface of the upper crust and the top interface of the middle crust of the velocity structure of the crust. It is very interesting that the Haicheng earthquake of Feb. 4, 1975,M 7.3 occurred at the depth gradient belt of the Curie isotherm surface. So, to research on the geomagnetic layers and the distribution characteristics of the Curie isotherm surface is meaningful in judging potential seismic foci. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica,15, 448–454, 1993. The data of velocity and electricity materials used in this paper are as reference from the related materials of the maps and directions etc. of Donggou-Haicheng-Dong Ujimqin Qi Geoscience Transect by Prof. Zao-Xun LU, Huai-Kuan XIA and others.     

The deepening of the wind-Mixed layer

Abstract

A simple model is given that describes the response of the upper ocean to an imposed wind stress. The stress drives both mean and turbulent flow near the surface, which is taken to mix thoroughly a layer of depth h, and to erode the stably stratified fluid below. A marginal stability criterion based on a Froude number is used to close the problem, and it is suggested that the mean momentum has a strong role in the mixing process. The initial deepening is predicted to obey

where u. is the friction velocity of the imposed stress, N the ambient buoyancy frequency, and t the time.

After one-half inertial period the deepening is arrested by rotadeon at a depth h = 2^2/4 u.{(Nf)⁺

where f is the Coriolis frequency. The flow is then a “mixed Ekman” layer, with strong inertial oscillations superimposed on it. Three quarters of the mean energy of the deepening layer is found to be kinetic, and only one-quarter potential.

Heating and cooling are included in the model, but stress dominates for time-scales of a day or less. Non-uniform stratification and currents existing prior to the onset of the wind are easily included.

Agreement between the first formula above and laboratory experiments of Kato and Phillips is very satisfactory; the second formula is consistent with observations of Francis and Stommel, though a more thorough test is needed. Oceanic observations in general support the assumption of slab-like mean profiles and direct response of the fluid to local winds.     

The equatorial dynamics of a shallow,homogeneous ocean

Abstract

It is shown that the linear equatorial dynamics of a shallow ocean is characterized by two boundary layers of width γ^? L and γL (γ is the Ekman number of the flow, assumed small, and L is a horizontal dimension of the basin). In the γ^? layer stress in the bottom Ekman layer is comparable to that in the surface Ekman layer. In the γ layer vertical friction is important throughout the depth of the ocean. Should the Rossby number ? be so large as to invalidate a linear theory (? > γ^5/3), then inertial effects become important at a distance ?^2/5 L from the equator. The role played in the circulation of the basin by the non-linear equatorial current first studied by Charney (1960) is shown to be similar to that of the γ layer of the linear theory. Though lateral friction is unimportant in a linear model of the flow, shear layers at the equator are found to be a necessary feature of non-linear flow.     

Introduction of the soil/vegetation/atmosphere continuum in a conceptual rainfall/runoff model

Abstract

This paper presents the inclusion of the soil-vegetation-atmosphere interactions in a proven conceptual model. This new scheme simulates the daily streamflows over small catchments by taking into account the average characteristics of the surface (soil and vegetation) for the calculation of actual evaporation and evapotranspiration. The model also simulates the daily evolution of soil moisture in two layers: the surface layer representing the first ten centimetres of the soil and the bulk layer representing the root zone. The results of the model calibration on a test site, and the results of the model validation on 36 watersheds, show its good capability to simulate streamflows and soil moisture in the surface layer and in the bulk soil layer. These first results are very encouraging and open the possibility of using these quantities for hydrological applications.     

Field investigation of a dry detention pond with underground detention storage

Abstract

A dry pond is an urban drainage component designed to temporarily store stormwater runoff and to encourage infiltration of surface water to the subsurface layer. This paper investigates field measurement of a dry pond at Taiping Health Clinic, Perak, Malaysia that has been functioning well for five years. The pond has a surface area of 195 m², maximum depth of 32 cm, and a storage capacity of 31.88 m³. The study focused on the infiltration functionality of the constructed dry pond and the results show that it has an average infiltration rate of 125 mm/h and dries up in 330 min after being filled to a depth of 31 mm. A public-domain hydrological model was then employed to simulate hydrographs of ponding and draining, the results of which matched observations with 86–98% accuracy. These results can lead to better understanding of the system and allow duplication of such a drainage design elsewhere.

Editor D. Koutsoyiannis

Citation Lai, S.H. and Mah, D.Y.S., 2012. Field investigation of a dry detention pond with underground detention storage. Hydrological Sciences Journal, 57 (6), 1249–1255.     

The incidence of internal waves onto a thin submerged barrier

Abstract

The problem of oblique incidence of internal ocean waves on a thin submerged ocean barrier is considered when the ocean has exponential density stratification. A Wiener-Hopf approach is used combined with numerical evaluation of series. Results for the reflected energy are obtained and reveal a complex dependence on incidence and barrier height. Application of this model to waves incident on the Mid-Atlantic ridge suggests that the ridge almosts isolates first mode energy on one side of the ocean from the other side. In certain circumstances there, is a surprising appearance of “barrier” waves. These waves are closely confined to the barrier and propagate along it.     

Several effects of a baroclinic current on the cross‐stream propagation of inertial‐internal waves†

Several effects of a baroclinic current on inertial‐internal waves at constant frequency are investigated, primarily through use of the method of characteristics. The special case of waves propagating transverse to a baroclinic current is considered. When the slope of an isopycnal is of the same order of magnitude as the slope of the characteristics, appreciable asymmetries are induced in the characteristics, the phase and group velocities, and the solution itself. These asymmetric effects are especially significant for waves at the low frequency end of the passband for free waves. Also, modifications occur to the passband, resulting in anomalously high and low frequency bands. The effective local inertial frequency, σ_f = [f(f+v_x )]^1/2, separates the normal and anomalously low frequency bands. Hence, the low frequency limit of the normal frequency band increases or decreases depending upon whether the horizontal shear in the mean flow is cyclonic or anticyclonic. In the anomalous frequency bands, the slopes of both characteristics have the same sign, causing various refraction and reflection phenomena. If the absolute value of the slope, s, of an isopycnal exceeds its critical value, s_c = effective local inertial frequency/Väisälä‐Brunt frequency, the anomalously low frequency band extends to imaginary frequencies. If s ? 0, the reflection of waves from a boundary is modified, the effective wavelength is increased, and the lines of constant phase are tilted from the vertical. For the general solution, discontinuities in the first‐order partial derivatives of the velocity field occur across certain characteristics. The nonseparable normal modes do not exhibit these discontinuous derivatives, but they only satisfy one of the two pairs of kinematic boundary conditions in rectangular regions.     

A numerical study of the string function using a primitive equation ocean model

Abstract

We use results from a primitive-equation ocean numerical model (SCRUM) to test a theoretical 'string function' formulation put forward by Tyler and Käse in another article in this issue. The string function acts as a stream function for the large-scale potential energy flow under the combined beta and topographic effects. The model results verify that large-scale anomalies propagate along the string function contours with a speed correctly given by the cross-string gradient. For anomalies having a scale similar to the Rossby radius, material rates of change in the layer mass following the string velocity are balanced by material rates of change in relative vorticity following the flow velocity. It is shown that large-amplitude anomalies can be generated when wind stress is resonant with the string function configuration.     

Baroclinic instability with variable static stability—a design study for a spherical atmospheric model experiment

Abstract

Exact solutions are obtained for a quasi-geostrophic baroclinic stability problem in which the rotational Froude number (inverse Burger number) is a linear function of the height. The primary motivation for this work was to investigate the effect of a radially-variable, dielectric body force, analogous to gravity, on baroclinic instability for the design of a spherical, synoptic-scale, atmospheric model experiment for a Spacelab flight. Such an experiment cannot be realized in a laboratory on the Earth's surface because the body force cannot be made strong enough to dominate terrestrial gravity. Flow in a rotating, rectilinear channel with a vertically variable body force and with no horizontal shear of the basic state is considered. The horizontal and vertical temperature gradients of the basic and reference states are taken as constants. Consequences of the body force variation and the other assumptions of the model are that the static stability (Brunt-Väisälä frequency squared) and the vertical shear of the basic state flow have the same functional form and that the transverse gradient of the potential vorticity of the basic state vanishes. The solutions show that the stability characteristics of the model are qualitatively similar to those of Eady's model. A short wavelength cutoff and a wavenumber of maximum growth rate are present. Further, the stability characteristics are quantitatively similar to Eady's results for parameters based on the vertically averaged Brunt-Väisälä: frequency. The solutions also show that the temperature amplitude distribution is particularly sensitive to the vertical variation of the static stability. For the static stability and shear decreasing (increasing) with height a relative enhancement of the temperature amplitude occurs at the lower (upper) surface. The other amplitudes and phases are only slightly influenced by the variation. The implication for the Spacelab experiment is that the variable body force will not significantly alter the dynamics from the constant gravity case. The solutions can be relevant to other geophysical fluid flows, including the atmosphere, ocean and annulus system in which the static stability undergoes variation with height.     

Ra in the Dead Sea

The²²⁸Ra concentrations of the Dead Sea waters range from 0.13 to 1.48 dpm kg⁻¹, two to three orders of magnitude higher than those of ocean waters and lake waters. However, the²²⁸Ra/²²⁶Ra activity ratios, (0.12–1.29) × 10⁻², are in the range reported for the hydrosphere.The surface waters of the Dead Sea are enriched in²²⁸Ra by a factor of about three over the near-bottom waters. There is a factor of about two spatial variability in the mid-depth Ra concentrations at the two profile stations. The near-bottom²²⁸Ra gradients yield vertical eddy diffusivity coefficient (K) of 2.0 and 0.4 cm² s⁻¹ at profile locations 1 and 2 respectively. These values are comparable to those measured in oceans and lakes.     

Time of concentration: a paradox in modern hydrology

Abstract

The time of concentration is a primary parameter for a variety of modern hydrological models adopted in professional and scientific communities. Nevertheless, a universally accepted working definition of this parameter is currently lacking and several definitions can be found in the technical literature along with related estimation procedures. This study brings to light the inherent variability of these definitions through the empirical analysis of four small basins. These case studies demonstrate that available approaches for the estimation of the time of concentration may yield numerical predictions that differ from each other by up to 500%.

Editor D. Koutsoyiannis

Citation Grimaldi, S., Petroselli, A., Tauro, F. and Porfiri, M., 2012. Time of concentration: a paradox in modern hydrology. Hydrological Sciences Journal, 57 (2), 217–228.     

The role of the helicity spectrum function in turbulent dynamo theory

Abstract

It is shown that, for general homogeneous turbulence, the anti-symmetric part of the spectrum tensor can be expressed in terms of a single scalar function H(k,ω) (the helicity spectrum function). Under the first-order smoothing approximation, the coefficients α _ij β _ijk in the expansion of the mean electromotive force in terms of the mean magnetic field are determined; α _ij is a weighted integral of H(k,ω), and β _ijk contains a part β(a)ijk which is likewise a weighted integral of H(k, ω). When the turbulence is axisymmetric, β(a)ijk contains Rädler's (1969a) “Ω ∧ J-effect”. It is shown that when the turbulence is statistically symmetric about a plane perpendicular to the axis of symmetry, then βij = O but the Rädler effect is non-zero. Explicit expressions for αij and βijk are given when the velocity field is generated by random forcing in a rotating medium. Finally, it is shown by means of a local analysis that the Rädler effect, in conjunction with uniform mean shear, can give rise to non-oscillatory dynamo action, and it is argued that this effect may be significant in the well-mixed interior of a stellar convection zone, where by symmetry the α-effect may be weak.