Conserved Properties

For long term simulations, it turns out that the reproduction of the conservation properties is most important in order to ensure reliable results. 

Beneath the conservation properties of QCMD its equations of motion possess another important geometric structure by being time reversible. As shown in [10], the application of symmotric integrators to reversible problems yields... 

For long-term simulations, it generally proves advantageous to consider numerical integrators which pass the structural properties of the model onto the calculated solutions. Hence, a careful analysis of the conservation properties of QCMD model is required. A particularly relevant constant of motion of the QCMD model is the total energy of the system... 

In the context of this paper, the most important conservation property of QCMD is related to its canonical Hamiltonian structure which implies the symplecticncs.s of the solution operator [1]. There are different ways to... 

Evaporation concentrates the dissolved constituents of seawater. Because the assumed inflow is twice the assumed outflow, conservative properties—properties that are not affected by precipitation, dissolution, or exchange with the atmosphere—are concentrated by a factor of 2. This increase in concentration changes the balance between the dissolved car-... 

A conservative property is at steady-state when fluxes, sources, and sinks do not change with time. It is not to be confused with equilibrium which is a state with no flux, no source, and no sink. The general transport equation (8.4.3) of element i at steady-state is... 

The mathematical models used to infer rates of water motion from the conservative properties and biogeochemical rates from nonconservative ones were flrst developed in the 1960s. Although they require acceptance of several assumptions, these models represent an elegant approach to obtaining rate information from easily measured constituents in seawater, such as salinity and the concentrations of the nonconservative chemical of interest. These models use an Eulerian approach. That is, they look at how a conservative property, such as the concentration of a conservative solute C, varies over time in an infinitesimally small volume of the ocean. Since C is conservative, its concentrations can only be altered by water transport, either via advection and/or turbulent mixing. Both processes can move water through any or all of the three dimensions... 

The formation of a shock wave is dependent on the objects that affect the flow field. The conservation of mass, momentum, and energy must be satisfied at any location. This is manifested in the formation of a shock wave at a certain location in the flow field to meet the conservahon equations. In the case of a blunt body in a supersonic flow, the pressure increases in front of the body. The increased pressure generates a detached shock wave to satisfy the conservation equations in the flow field to match the conserved properties between the inflow and outflow in front of the body. The velocity then becomes a subsonic flow behind the detached shock wave. However, the shock wave distant from the blunt body is less affected and the detached shock wave becomes an oblique shock wave. Thus, the shock wave appears to be curved in shape, and is termed a bow shock wave, as illustrated in Fig. C-1. 

This discretization method obeys a conservation property, and therefore is called conservative. With the exception of the first element and the last element, every element face is a part of two elements. The areas of the coincident faces and the forces on them are computed in exactly the same way (except possibly for sign). Note that the sign conventions for the directions of the positive stresses is important in this regard. The force on the left face of some element is equal and opposite to the force on the right face of its leftward neighbor. Therefore, when the net forces are summed across all the elements, there is exact cancellation except for the first and last elements. For this reason no spurious forces can enter the system through the numerical discretization itself. The net force on the system of elements must be the net force caused by the boundary conditions on the left face of the first element and the right face of the last element. 

Vary the coefficients of the feedback matrix in the reference filter while maintaining the energy conserving property, or similarly vary the allpass gains of an allpass feedback loop reverberator. 

Another conserved property is the total energy, and in terms of local energy density pe for each point in the system, we have... 

At stationary state, all the properties of a system are independent of temperature. Stationary states resemble equilibrium states in their invariance with time however, they differ in that flows still continue to occur and entropy is produced in the system. If a property is conservative, then the divergence of the corresponding flow must vanish for example, dp/dt = div J. Therefore, the steady flow of a conservative quantity must be source-free and in stationary states the flows of conservative properties are constant. 

With these specifications, and with the appropriate neutral particle-plasma collision terms put into the combined set of neutral and plasma equations, internal consistency within the system of equations is achieved. Overall particle, momentum and energy conservation properties in the combined model result from the symmetry properties of the transition probabilities W indices of pre-collision states may be permuted, as well as indices of postcollision states. For elastic collisions even pre- and post collision states may be exchanged in W. 

The reason for the inequality sign is that entropy is not a conserved property. In fact, entropy is produced whenever spontaneous processes occur such as the irreversible flow of heat across a finite temperature differential. Only in the hypothetical case when heat is transferred reversibly to a body at constant temperature will the entropy increase equal the heat addition divided by the absolute temperature. 

Below the thermocline, the temperature changes only little with depth. The temperature is a non-conservative property of seawater because adiabatic compression causes a slight increase in the in situ temperature measured at depth. For instance in the Mindanao Trench in the Pacific Ocean, the temperature at 8500 and 10,000 m is 2.23 and 2.48 °C, respectively. The term potential temperature is defined to be the temperature that the water parcel would have if raised adiabatically to the ocean surface. For the examples above, the potential temperatures are 1.22 and 1.16 °C, respectively. Potential temperature of seawater is a conservative index. 

When the conservable property is represented by the local quantity of the species A (P = UjnA, r = = Cft) transported by molecular and convective... 

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