Advective derivative

To motivate the geostrophic approximation, we invoke a scale analysis approach. Assume frictional forces can be neglected, and the atmospheric motions have a characteristic horizontal length-scale, L, and velocity scale, U. Recalling the definition of the advective derivative operator D/Dt, we find that the magnitude of the acceleration terms, Dw/Dt and Du/Dt in Eqs. (9.2.20) and (9.2.21), is 17 /L, provided the magnitude of the time scale of the motion is greater than or equal to the advective time, L/U. The terms proportional to tan 0 are of order U /a, and the Coriolis terms are of order /(/. If L < a, then the ratio of each of the terms to the Coriolis term is less than or comparable to the Rossby number, defined as... 

For large-scale motion with a time scale greater than a Martian day, only the term wdT/dz contributes significantly to the advective derivative of the temperature, DT/Dt, on the left side of Eq. (9.2.24). Wth this approximation and Eq. (9.2.31), the thermodynamic energy equation reduces to the relatively simple relation... 

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

Figure 7. Sediment eontains derived both seavenged from the water eolumn during particle settling and contained in solid material. Ra produced in the sediments is highly soluble in pore waters and diffuses into the overlying water or is advected across the sediment-water interface by discharging groundwater. Rn is produced within the water column from dissolved Ra and within the underlying sediments.

In this paper we present for the first time a test that combines heat extraction and heat injection pulses in one experiment. It is expected that differences in the ground thermal conductivity, when different data windows are used to obtain an estimate, can be related to advection and convection of ground water. The real ground conductivity should be derived from the experimental data where the response is close to or lower than the natural ground temperature, minimizing effects of advection and convection. First results, for a case of no ground water flow, show that estimates of ground thermal conductivity are very comparable for the different data windows. 

Truncation error arises from approximating each of the various space and time derivatives in the transport equation. The error resulting from the derivative in the advection term is especially notable and has its own name. It is known as numerical dispersion because it manifests itself in the calculation results in the same way as hydrodynamic dispersion. 

When the Pecet number, the measure of the relative importance of advection to diffusion, is small, which is the case for high viscosity magmas, the temporal derivation of concentration and the advection term in Eq. (13.31) may be ignored, and quasi-static approximation may be developed. In this case, Eq. (13.31) reduces to... 

Recall the distinction between advective and diffusive transport, which we made in Section 18.1 while traveling in the dining car through the Swiss Alps. We then introduced Fick s first law to describe the mass flux per unit area and time by diffusion or by any other random process (Eq. 18-6). Rewritten in terms of partial derivatives, the diffusive flux along the x-axis is ... 

We will now discuss the steady-state solutions of Eq. 22-6. Remember that steady-state does not mean that all individual processes (diffusion, advection, reaction) are zero, but that their combined effect is such that at every location along the x-axis the concentration C remains constant. Thus, the left-hand side of Eq. 22-6 is zero. Since at steady-state time no longer matters, we can simplify C(x,t) to C(x) and replace the partial derivatives on the right-hand side of Eq. 22-6 by ordinary ones ... 

The model (Fig. 23.6) consists of three compartments, (a) the surface mixed water layer (SMWL) or epilimnion, (b) the remaining open water column (OP), and (c) the surface mixed sediment layer (SMSL). SMWL and OP are assumed to be completely mixed their mass balance equations correspond to the expressions derived in Box 23.1, although the different terms are not necessarily linear. The open water column is modeled as a spatially continuous system described by a diffusion/advection/ reaction... 

The corresponding dynamic equations of the open water column are constructed from Eq. 22-6. They are completed by the sediment-water boundary flux derived in Eq. 23-38. We assume that the net vertical advection of water is zero. Thus, the vertical water movement is incorporated in the turbulent diffusivity, Ez. The assumption implies that if chemicals are directly introduced at depth z (term 1), they would not be accompanied by significant quantities of water. Typically, such inputs are due to sewage outlets (treated or untreated) into the lake. We get ... 

For porous media it is convenient to choose C as the key variable since it is this concentration that is determined in filtered water samples taken at a well. The dynamic equation for Cw is derived from Eq. 25-10 according to the same procedure as in Box 18.5. Note that now both transport by advection and by diffusion are reduced by the relative fraction in dissolved form,/w ... 

In this form one sees an analogy in the vorticity equation to the other transport equations— a substantial-derivative description of advective transport, a Laplacian describing the diffusive transport, and possibly a source term. It is interesting to observe that the vorticity equation does not involve the pressure. Since pressure always exerts a normal force that acts through the center of mass of a fluid packet (control volume), it cannot alter the rotation rate of the fluid. That is, pressure variations cannot cause a change in the vorticity of a flow field. 

Begin by noting that C() satisfy the Neumann problem given by (5) and boundary conditions whose solution is Cj(x, y, t) = Cj(x, t). We now derive the overall macroscopic mass balance for the species. To this end we begin by deriving the closure problem for Cj. Given Cf(x,t), combine (6) with boundary conditions and neglect the advection induced by du/dt, to obtain the local Neumann problem... 

The magnitude of the artifact is such that coastal and wetland data in the literature is not likely to be seriously affected. An example of this is our Bahamas data ( ) which illustrates the dominance of local island sources elevated concentrations were preceded by wind shifts bringing air from the island of Andros which has extensive tidal flats. Similarly, the H2S concentrations measured on the Gulf of Mexico cruise in the same paper most likely reflect the advection of continental and tidally derived HiS. Brief periods of low H2S concentrations (< 10 ppt) were obtained in the Gulf Stream near the end of both of these cruises during brief exposure to easterly open ocean air. 

Advection is derived from the Latin words ad and vehere, meaning to carry to. Convection, used in the chemical engineering literature, derives from the Latin words con and vehere, meaning to bring together. In chaotic mixing, as indicated later in the chapter, the former appears to be the more appropriate term. 

Figure 1. Plot of the scavenging rate constants derived from advection-diffusion--scavenging models against the stability constant for simple hydroxo complexes, Ki...

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