On the steric and mass-induced contributions to the annual sea level variations in the Mediterranean Sea

The sea level variation (SLVtotal) is the sum of two major contributions: steric and mass-induced. The steric SLVsteric is that resulting from the thermal and salinity changes in a given water column. It only involves volume change, hence has no gravitational effect. The mass-induced SLVmass, on the other hand, arises from adding or subtracting water mass to or from the water column and has direct gravitational signature. We examine the closure of the seasonal SLV budget and estimate the relative importance of the two contributions in the Mediterranean Sea as a function of time. We use ocean altimetry data (from TOPEX/Poseidon, Jason 1, ERS, and ENVISAT missions) to estimate SLVtotal, temperature, and salinity data (from the Estimating the Circulation and Climate of the Ocean ocean model) to estimate SLVsteric, and time variable gravity data (from Gravity Recovery and Climate Experiment (GRACE) Project, April 2002 to July 2004) to estimate SLVmass. We find that the annual cycle of SLVtotal in the Mediterranean is mainly driven by SLVsteric but moderately offset by SLVmass. The agreement between the seasonal SLVmass estimations from SLVtotal – SLVsteric and from GRACE is quite remarkable; the annual cycle reaches the maximum value in mid-February, almost half a cycle later than SLVtotal or SLVsteric, which peak by mid-October and mid-September, respectively. Thus, when sea level is rising (falling), the Mediterranean Sea is actually losing (gaining) mass. Furthermore, as SLVmass is balanced by vertical (precipitation minus evaporation, P–E) and horizontal (exchange of water with the Atlantic, Black Sea, and river runoff) mass fluxes, we compared it with the P–E determined from meteorological data to estimate the annual cycle of the horizontal flux.

Reply to comment by L. Fenoglio-Marc et al. on “On the steric and mass-induced contributions to the annual sea level variations in the Mediterranean Sea”

Reply to comment by L. Fenoglio-Marc et al. on “On the steric and mass-induced contributions to the annual sea level variations in the Mediterranean Sea”.

On some solutions of a functional equation related to the partial sums of the Riemann zeta function

In this paper, we prove that infinite-dimensional vector spaces of α-dense curves are generated by means of the functional equations f(x)+f(2x)+⋯+f(nx)=0, with n≥2, which are related to the partial sums of the Riemann zeta function. These curves α-densify a large class of compact sets of the plane for arbitrary small α, extending the known result that this holds for the cases n=2,3. Finally, we prove the existence of a family of solutions of such functional equation which has the property of quadrature in the compact that densifies, that is, the product of the length of the curve by the nth power of the density approaches the Jordan content of the compact set which the curve densifies.

From the Farkas Lemma to the Hahn–Banach Theorem

This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) ≥ 0 which are consequences of a composite convex inequality (S ◦ g)(x) ≤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of f, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as equivalent to an extended version of the so-called Hahn–Banach–Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn–Banach theorem and the Mazur–Orlicz theorem for extended sublinear functions.