Penetrative convection in rotating fluids: A model for the base of stellar convection zones

Abstract

To model penetrative convection at the base of a stellar convection zone we consider two plane parallel, co-rotating Boussinesq layers coupled at their fluid interface. The system is such that the upper layer is unstable to convection while the lower is stable. Following the method of Kondo and Unno (1982, 1983) we calculate critical Rayleigh numbers R_c for a wide class of parameters. Here, R_c is typically much less than in the case of a single layer, although the scaling R_c～T^2/3 as T → ∞ still holds, where T is the usual Taylor number. With parameters relevant to the Sun the helicity profile is discontinuous at the interface, and dominated by a large peak in a thin boundary layer beneath the convecting region. In reality the distribution is continuous, but the sharp transition associated with a rapid decline in the effective viscosity in the overshoot region is approximated by a discontinuity here. This source of helicity and its relation to an alpha effect in a mean-field dynamo is especially relevant since it is a generally held view that the overshoot region is the location of magnetic field generation in the Sun.     

Turbulent transport of magnetic fields. V. Distribution of magnetic energy in a simple α2-dynamo

Abstract

In this paper we analyse the stationary mean energy density tensor T_ij = B_iB_j for the x ²-sphere. This model is one of the simplest possible turbulent dynamos, originally due to Krause and Steenbeck (1967): a conducting sphere of radius R with homogeneous, isotropic and stationary turbulent convection, no differential rotation and negligible resistivity. The stationary solution of the (linear) equation for T_ij is found analytically. Only T_rr , T _θθ and T _φφ are unequal to zero, and we present their dependence on the radial distance r.

The stationary solution depends on two coefficients describing the turbulent state: the diffusion coefficient β≈?u²?τ_c/3 and the vorticity coefficient γ ≈ ?|?×u|²?τ_c/3 where u(r, t) is the turbulent velocity and _c its correlation time. But the solution is independent of the dynamo coefficient α≈??u·?×u?τ_c/3 although α does occur in the equation for T_ij . This result confirms earlier conclusions that helicity is not required for magnetic field generation. In the stationary state, magnetic energy is generated by the vorticity and transported to the boundary, where it escapes at the same rate. The solution presented contains one free parameter that is connected with the distribution of B over spatial scales at the boundary, about which T_ij gives no information. We regard this investigation as a first step towards the analysis of more complicated, solar-type dynamos.     

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.     

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.     

Excess222Rn and the benthic boundary layer in the western and southern Indian Ocean

We present 9 bottom²²²Rn profiles measured from the western and southern Indian Ocean during the 1977–1978 GEOSECS expedition. These profiles can be grouped into three cypes: one-layer, two-layer, and irregular types. The one-layer profiles with quasi-exponential distributions allow one to estimate the apparent vertical eddy diffusivity,K_v, with a simple model. The two-layer profiles show that there is a benthic boundary layer of the order of 50–100 m in which the excess²²²Rn distribution shows a vertical gradient much smaller than that of the layer immediately above. Within the boundary layer, the STD potential temperature (θ) and density(σ₄) profiles are practically constant, and theK_v values are of the order of 1000 cm²/s. The STD profiles for the water column above the boundary layer show gradients of increasing stability, and theK_v values are of the order of 100 cm²/s. Modeling of the Rn data in the water column above the boundary layer indicates that there is a transition layer which effectively reduces the penetration of excess Rn from the benthic boundary layer into the upper layer.Sarmiento et al. [10] have shown that the buoyancy gradient or stability is inversely correlated with the apparent vertical eddy diffusivity, and the resulting buoyancy flux is fairly uniform, ranging from 1 to 14 × 10^?6 cm²/s³ in the Atlantic and Pacific Oceans. However, Sarmiento et al. [11] show that a much higher buoyancy flux is associated with an intensified flow of the bottom water through a passage. In the Indian Ocean basins, we have found that the buoyancy flux has a comparable range (3–14 × 10^?6 cm²/s³), except for a couple of stations where both stability and apparent vertical diffusivity are higher, resulting in a much higher buoyancy flux, probably indicative of rapid bottom water flow.     

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.     

Consequences of coastal upwelling events on physical and chemical patterns in the central Gulf of Finland (Baltic Sea)

The consequences of a coastal upwelling event on physical and chemical patterns were studied in the central Gulf of Finland. Weekly mapping of hydrographical and -chemical fields were carried out across the Gulf between Tallinn and Helsinki in July–August 2006. In each survey, vertical profiles of temperature and salinity were recorded at 27 stations and water samples for chemical analyses (PO₄³⁻, NO₂⁻+NO₃⁻) were collected at 14 stations along the transect. An ordinary distribution of hydrophysical and -chemical variables with the seasonal thermocline at the depths of 10–20 m was observed in the beginning of the measurements in July. Nutrient concentrations in the upper mixed layer were below the detection limit and nutriclines were located just below or in the lower part of the thermocline. In the first half of August, a very intense upwelling event occurred near the southern coast of the Gulf when waters with low temperature and high salinity from the intermediate layer surfaced. High nutrient concentrations were measured in the upwelled water – 0.4 μmol l⁻¹ of phosphates and 0.6 μmol l⁻¹ of nitrates+nitrites. We estimated the amount of nutrients transported into the surface layer as 238–290 tons of phosphorus (P)-PO₄³⁻ and 175–255 tons of N-NO_x for a 12 m thick, 20 km wide and 100 km long coastal stretch. Taking into account a characteristic along-shore extension of the upwelling of 200 km, the phosphate-phosphorus amount is approximately equal to the average total monthly riverine load of phosphorus to the Gulf of Finland. It is shown that TS-characteristics of water masses and vertical distribution of nutrients along the study transect experienced drastic changes caused by the upwelling event in the entire studied water column. TS-analysis of profiles obtained before and during the upwelling event suggests that while welled up, the cold intermediate layer water was mixed with the water from the upper mixed layer with a share of 85% and 15%. We suggest that the coastal upwelling events contribute remarkably to the vertical mixing of waters in the Gulf of Finland. Intrusions of nutrient-rich waters along the inclined isopycnal surfaces in the vicinity of upwelling front were revealed. The upwelling event widened the separation of phosphocline and nitracline which in turn prevented surfacing of nitrate+nitrite-nitrogen during the next upwelling event observed a week after the upwelling relaxation. A suggestion is made that such widening of nutricline separation caused by similar upwelling events in early summer could create favourable conditions for late summer cyanobacterial blooms.     

Non-axisymmetric α2Ω-dynamo waves in thin stellar shells

Linear α²Ω-dynamo waves are investigated in a thin turbulent, differentially rotating convective stellar shell. A simplified one-dimensional model is considered and an asymptotic solution constructed based on the small aspect ratio of the shell. In a previous paper Griffiths et al. (Griffiths, G.L., Bassom, A.P., Soward, A.M. and Kuzanyan, K.M., Nonlinear α²Ω-dynamo waves in stellar shells, Geophys. Astrophys. Fluid Dynam., 2001, 94, 85–133) considered the modulation of dynamo waves, linked to a latitudinal-dependent local α-effect and radial gradient of the zonal shear flow. These effects are measured at latitude θ by the magnetic Reynolds numbers R _α f(θ) and R _Ω g(θ). The modulated Parker wave, which propagates towards the equator, is localised at some mid-latitude θ_p under a Gaussian envelope. In this article, we include the influence of a latitudinal-dependent zonal flow possessing angular velocity Ω_*(θ) and consider the possibility of non-axisymmetric dynamo waves with azimuthal wave number m. We find that the critical dynamo number D _c?=?R _α R _Ω is minimised by axisymmetric modes in the αΩ-limit (Rα→0). On the other hand, when Rα?≠?0 there may exist a band of wave numbers 0?m?m ^? for which the non-axisymmetric modes have a smaller D _c than in the axisymmetric case. Here m ^? is regarded as a continuous function of R _α with the property m?→0 as R _α→0 and the band is only non-empty when m??>1, which happens for sufficiently large R _α. The preference for non-axisymmetric modes is possible because the wind-up of the non-axisymmetric structures can be compensated by phase mixing inherent to the α²Ω-dynamo. For parameter values resembling solar conditions, the Parker wave of maximum dynamo activity at latitude θ_p not only propagates equatorwards but also westwards relative to the local angular velocity Ω_*(θ _p ). Since the critical dynamo number D _c?=?R _α R _Ω is O (1) for small R _α, the condition m ^??>?1 for non-axisymmetric mode preference imposes an upper limit on the size of |dΩ_*/dθ|.     

Kinematic stationary geodynamo models with separated toroidal and poloidal motions

Abstract

This paper is concerned with a three-dimensional spherical model of a stationary dynamo that consists of a convective layer with a simple poloidal flow of the S^2c ₂ kind between a rotating inner body core and solid outer shell. The rotation of the inner core and the outer shell means that there are regions of concentrated shear or differential rotation at the convective layer boundaries. The induction equation for the inside of the convective layer was solved numerically by the Bullard-Gellman method, the eigenvalue of the problem being the magnetic Reynolds number of the poloidal flow (R _M2) and it was assumed that the magnetic Reynolds number of the core (R _M1) and of the shell (R _M3) were prescribed parameters. Hence R _M2 was studied as a function of R _M1 and R _M3, along with the orientation of the rotation axis, the radial dependence of the poloidal velocity and the relative thickness of the layers for the three different situations, (i) the core alone rotating, (ii) the shell alone rotating and (iii) the core and the shell rotating together. In all three cases it was found that, at definite orientations of the rotation axis, there is a good convergence of both the eigenvalues and the eigenfunctions of the problem as the number of spherical harmonics used to represent the problem increases. For R _M1 =R _M3= 10³, corresponding to the westward drift velocity and the parameters of the Earth's core, the critical values of R _M2 are found to be three orders of magnitude lower than R _M1, R _M3 so that the poloidal flow velocity sufficient for maintaining the dynamo process is 10-20 m/yr. With only the core or the shell rotating, the velocity field generally differs little from the axially symmetric case. However, for R _M2 (or R _M3) lying in the range 10² to 10⁵, the self-excitation condition is found to be of the form R _M2˙R ^½ _M1=constant (or R _M2˙R^½ _M3=constant) and the solution does not possess the properties of the Braginsky near-axisymmetric dynamo. We should expect this, in particular, in the Braginsky limit R _M2˙R^?½; _M1=constant.

An analysis of known three-dimensional dynamo models indicates the importance of the absence of mirror symmetry planes for the efficient generation of magnetic fields.     

Dynamics of seasonal recharge beneath a semiarid vegetation on the gnangara mound,Western Australia

To estimate seasonal changes in recharge to the underlying sandy aquifer, the soil water dynamics of the unsaturated zone was monitored down to a depth of 20 m over a period of three years (1985 to 1987). The measurements were made by a neutron probe at eight locations beneath a native vegetation in a semiarid region, Western Australia, receiving precipitation of 775 mm yr^?1. A relatively simple method, based on the analyses of sequentially measured soil water profiles involving utilization of zero flux plane in the unsaturated zone, is presented and used to compute seasonal recharge rates. Drainage fluxes (recharge rates) below two specified depths were estimated. These were: R₁ (water flux at a depth of 10 m, just below the maximum rooting depth) and R₂ (water flux at a depth of 18 m, just above the water table). These two estimates were significantly different both on a seasonal and annual basis, but their cumulative values for the three year period were very similar. While the annual precipitation varied from 525 to 850 mm yr^?1, the corresponding spatially averaged R₁ varied from 34 to 149 mm yr^?1, and R₂ varied from 65 to 80 mm yr^?1. A significant difference in recharge between the upslope and downslope positions on a hillslope was ascribed to differences in vegetation density of the understorey and differences in hydraulic properties of subsoils. For the three year period, the average R₁ and R₂ were 13 per cent and 10 per cent of the precipitation respectively. These values compare favourably with a long-term estimate based on an environmental tracer technique.     

Transitions to chaotic thermal convection in a rapidly rotating spherical fluid shell

Abstract

Numerical simulations of thermal convection in a rapidly rotating spherical fluid shell heated from below and within have been carried out with a nonlinear, three-dimensional, time-dependent pseudospectral code. The investigated phenomena include the sequence of transitions to chaos and the differential mean zonal rotation. At the fixed Taylor number T _a =10⁶ and Prandtl number Pr=1 and with increasing Rayleigh number R, convection undergoes a series of bifurcations from onset of steadily propagating motions SP at R=R _c = 13050, to a periodic state P, and thence to a quasi-periodic state QP and a non-periodic or chaotic state NP. Examples of SP, P, QP, and NP solutions are obtained at R = 1.3R _c , R = 1.7 R _c , R = 2R _c , and R = 5 R _c , respectively. In the SP state, convection rolls propagate at a constant longitudinal phase velocity that is slower than that obtained from the linear calculation at the onset of instability. The P state, characterized by a single frequency and its harmonics, has a two-layer cellular structure in radius. Convection rolls near the upper and lower surfaces of the spherical shell both propagate in a prograde sense with respect to the rotation of the reference frame. The outer convection rolls propagate faster than those near the inner shell. The physical mechanism responsible for the time-periodic oscillations is the differential shear of the convection cells due to the mean zonal flow. Meridional transport of zonal momentum by the convection cells in turn supports the mean zonal differential rotation. In the QP state, the longitudinal wave number m of the convection pattern oscillates among m = 3,4,5, and 6; the convection pattern near the outer shell has larger m than that near the inner shell. Radial motions are very weak in the polar regions. The convection pattern also shifts in m for the NP state at R = 5R _c , whose power spectrum is characterized by broadened peaks and broadband background noise. The convection pattern near the outer shell propagates prograde, while the pattern near the inner shell propagates retrograde with respect to the basic rotation. Convection cells exist in polar regions. There is a large variation in the vigor of individual convection cells. An example of a more vigorously convecting chaotic state is obtained at R = 50R _c . At this Rayleigh number some of the convection rolls have axes perpendicular to the axis of the basic rotation, indicating a partial relaxation of the rotational constraint. There are strong convective motions in the polar regions. The longitudinally averaged mean zonal flow has an equatorial superrotation and a high latitude subrotation for all cases except R = 50R _c , at this highest Rayleigh number, the mean zonal flow pattern is completely reversed, opposite to the solar differential rotation pattern.     

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 .     

A numerical model for estimating groundwater flux from subsurface temperature profiles

A simple numerical model is presented for estimating vertical groundwater flux from transient subsurface temperature profiles obtained from field measurements. The model developed utilizes the MacCormack scheme, which is based on the Finite Difference Method (FDM), for solving the governing partial differential equation of convection–diffusion heat transport with appropriate initial and boundary conditions within the subsurface. In order to validate the model, numerical solutions obtained for the study area located in the Nagoka plain, Japan are compared with the published measured data and results obtained by others. Results obtained show good agreement and fit the observed data with a correlation coefficient, R², of 0·88. The estimated groundwater flux is 1·85 × 10⁻⁷ m s⁻¹. Sensitivity analyses were also carried out to investigate the effect of variations in groundwater fluxes, thermal properties and the annual thermal variability due to climatic changes on the transient subsurface temperature profiles and to have a better understanding of the subsurface thermal dynamics. A substantial effect of annual climatic variability is observed on the temporal distributions of temperature depth profiles, and a better estimate of thermal parameters is required to estimate vertical groundwater flux. The largest change in subsurface temperature depth profiles due to groundwater flux over a year is within ± 4 °C. The influence of groundwater flux on subsurface temperature distributions in space and time may be more pronounced in areas where the top of the saturated layer fluctuates considerably. Variation in thermal diffusivity results in temperature change up to ± 1·5% and may cause change in groundwater flux estimate by ± 18%. The model presented has merits over analytical solutions (type curve matching techniques) in terms of suitability and applicability to real field problems, and can be a good asset to hydrological models as quantifying groundwater recharge or deducing it from other quantities, such as rainfall, evapotranspiration and runoff, is often complicated. Copyright © 2007 John Wiley & Sons, Ltd.     

Vertical mixing in Überlingersee (Lake Constance) traced by SF6 and heat

Vertical mixing in Überlingersee is studied by releasing sulfur hexafluoride (SF₆) as a tracer at a central hypolimnic depth of 60 m and measuring its subsequent vertical dispersion over a period of three months. The experiment started with a streaky tracer injection of 1 liter gaseous SF₆ (STP) in August 1990. At that time the lake showed a typical strong summer stratification which in a weakened form lasts until November. From the SF₆ profiles of fifteen surveys at three sampling sites vertical diffusivitiesK _z are calculated compensating internal seiche displacement and horizontal tracer loss. Except of the bottom region no sampling site or time period is marked by significant differences in the hypolimnicK _z profile. So vertical mixing in the whole Überlingersee is described by mean diffusivities decreasing from 1.7 cm²/s at 120 m depth to 0.4 cm²/s in 30 m. The minimal value of 0.3 cm²/s in the thermocline region at 20 m depth is only based on observations in autumn. For a strong summer stratification it is certainly lower. The gradient-flux-method for heat was applied to compute a meanK _z (T) profile from continuously measured temperature profiles. Significant differences resulting from the two tracers showed, that theK _z (T) values are underestimated by up to a factor of 5 if cooling by lateral exchange is neglected. Particularly, internal seiche pumping of colder water from the adjacent Lake Obersee over the separating sill of Mainau into the deep Überlingersee basin is observed in 1990 from August onward, obviously controlling the heat budget below the sill level.     

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.     

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.     

Pan evaporation modelling and changing attribution analysis on the Tibetan Plateau (1970–2012)

Pan evaporation (E_p) is an important indicator of water and energy and the decline of E_p has been reported in many regions over the last decades. The climate and E_p are dependent on each other. In this study, the temporal trends of E_p and main E_p drivers, namely mean air temperature (T_a), wind speed (u), global solar radiation (R_s), net long‐wave radiation(R_nl) and vapour pressure deficit (D) from 1970 to 2012, were calculated on the basis of 26 meteorological stations on the Tibetan Plateau. The arithmetic average of E_p from 26 stations decreased with the rate of ?11.91 mm a^?2; the trends of R_s, R_nl, T_a, u and D were ?1.434 w m^?2 decade^?1, 0.2511 w m^?2 decade^?1, 0.3590°C decade^?1, ?0.2376 m s^?1 decade^?1 and 9.523 Pa decade^?1, respectively. The diffuse irradiance is an essential parameter to model E_p and quantify the contribution of climatic factors to changing E_p. 60 724 observations of R_s and diffuse solar irradiance (R_d) from seven of the 26 stations were used to develop the correlation between the diffuse fraction (R_d/R_s), and the clearness index (R_s/R_o). On the basis of the estimation of the diffuse component of R_s and climatic data, we modified the PenPan model to estimate Chinese micro‐pan evaporation (E_p) and assess the attribution of E_p dynamics using partial derivatives. The results showed that there was a good agreement between the observed and calculated daily E_p values. The observed decrease in E_p was mostly due to declining wind speed (?13.7 mm a^?2) with some contributions from decreasing solar irradiance (?3.1 mm a^?2); and the increase of temperature had a large positive effect (4.55 mm a^?2) in total whilst the increase of R_nl had insignificant effect (0.35 mm a^?2) on E_p rates. The change of E_p is the net result of all the climatic variables. Copyright © 2014 John Wiley & Sons, Ltd.     

Modelling spatial variations in dissolved oxygen in fine‐grained streams under uncertainty

This paper presents a novel triple‐layer model, called VART DO‐3L, for simulation of spatial variations in dissolved oxygen (DO) in fine‐grained streams, characterized by a fluid mud (fluff or flocculent) layer (an advection‐dominated storage zone) as the interface between overlying stream water and relatively consolidated streambed sediment (a diffusion‐dominated storage zone). A global sensitivity analysis is conducted to investigate the sensitivity of VART DO‐3L model input parameters. Results of the sensitivity analysis indicate that the most sensitive parameter is the relative size of the advection‐dominated storage zones (A_s/A), followed by a lumped reaction term (R) for the flocculent layer, biological reaction rate (μ_o) in diffusive layer and biochemical oxygen demand concentration (L) in water column. In order to address uncertainty in model input parameters, Monte Carlo simulations are performed to sample parameter values and to produce various parameter combinations or cases. The VART DO‐3L model is applied to the Lower Amite River in Louisiana, USA, to simulate vertical and longitudinal variations in DO under the cases. In terms of longitudinal variation, the DO level decreases from 7.9 mg l at the Denham Springs station to about 2.89 mg l^?1 at the Port Vincent station. In terms of vertical variation, the DO level drops rapidly from the overlying water column to the advection‐dominated storage zone and further to the diffusive layer. The DO level (C_F) in the advective layer (flocculent layer) can reach as high as 40% of DO concentration (C) in the water column. The VART DO‐3L model may be applied to similar rivers for simulation of spatial variations in DO level. Copyright © 2014 John Wiley & Sons, Ltd.     

Excess bottom222Rn profiles and their implications in the northwestern Pacific Ocean

Vertical profiles of excess bottom²²²Rn and potential temperature were measured at 23 stations in the northwestern Pacific Ocean. The Rn profiles were classified into the following three types: quasi-exponential (type E), benthic boundary layer (type B), and horizontally disturbed (type H). The ratio among types E, B and H was approximately 2 : 3 : 1.An apparent vertical eddy diffusivity (K) was calculated by applying a one-dimensional diffusion model to the Rn profiles of types E and B. Type E had K values ranging from 15 to 180 cm² s^?1 (average: 70 cm² s^?1). As to type B, K values for the benthic boundary layer (4.5–260 cm² s^?1, average: 120 cm² s^?1) were always more than an order of magnitude larger than those for the upper layer (0.2–35 cm² s^?1, average: 7 cm² s^?1), indicating more active vertical mixing in the benthic boundary layer than in the upper layer.Rn profiles were measured in regions where the bottom topography is known. It was verified that the occurrence of type H related closely with local bottom topographic features accompanied by lateral transient supply of Rn-rich or Rn-poor water.A couple of Rn profiles at the same location, measured at time intervals of several years, were compared with each other for three locations. The general characteristics of Rn profiles were shown to remain unaltered with time, while the fine structure of Rn profiles may have short-term variations caused by local bottom topography and fluctuations of bottom current as indicated in type H.