Hydrostatic equation

The atmosphere is very close to being in hydrostatic equilibrium in the vertical dimension. This can be described by the hydrostatic equation  

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The choices come in defining M for this mixture of gases. We might define M, for each gas separately, or we might define a mean value M = 2, X, M where X, is the mole fraction of component /. The use of M, would hold in the absence of any physical mixing (e.g. by turbulence or Brownian motion), while M would be used in the case of perfect mixing. 

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However, away from the surface, processes frequently are adiabatic. For example, if a volume (parcel) of air is forced upward over a ridge, the upward-moving air will encounter decreased atmospheric pressure and will expand and cool. If the air is not saturated with water vapor, the process is called dry adiabatic. Since no heat is added or subtracted. Ah in Eq. (17-13) can be set equal to zero, and introducing the hydrostatic equation... 

The total flow rate, (Lmn + Gmn), leaving the mixer will be related to the total phase volumes VLm and Vcm by a hydrostatic equation, which will depend on the net difference in the head of liquid, between the levels in the mixer and in the settler. The actual form of this relationship might need to be determined experimentally, but could, for example, follow a simple, square-root relationship of the form, in which flow rate is proportional to the square root of the difference in liquid head, or indeed to the total volume of liquid in the mixer, e.g.. 

Once again, using the idea of a homology transformation, we can approximate the hydrostatic equation (5.8) replacing dP/dm with PJM, leading to... 

Applying the homology version of the hydrostatic equation (5.8) to the envelope, one obtains another condition for the pressure at the edge of the core ... 

In order to calculate the injection pressure, one integrates the differential hydrostatic equation from the reservoir conditions to the surface. The hydrostatic equation is ... 

Unlike gases, liquids are typically quite incompressible. That is, their densities can be assumed to be constant. The integration of the hydrostatic equation is simple. The result is ... 

In order to estimate the injection pressure for a supercritical fluid, the hydrostatic equation must be carefully integrated - there are no short cuts. 

Atmospheric GCMs simulate the time evolution of various atmospheric fields (wind speed, temperature, surface pressure, and specific humidity), discretized over the globe, through the integration of the basic physical equations the hydrostatic equation of motion, the thermodynamic equation of state, the mass continuity equation, and a water vapor transport equation. To reproduce the... 

For constant density, p is constant and the above equation may be integrated to give the hydrostatic equation... 

Since the density of air is effectively zero, the contribution of the air to the 3-ft manometer reading can be neglected. The contribution to the pressure due to the carbon tetrachloride in the manometer is found by using the hydrostatic equation. 

The contribution to the pressure due to the height of water above the pressure gauge is similarly calculated using the hydrostatic equation. 

For part 2, application of the hydrostatic equation to determine gives... 

Besides the immediate consequence that the lifetime of radiation dominated stars becomes independent of the mass, as the amount of fuel is proportional to the rate at which they spend it, this obviously topples the premise in the flawed argument above that L grows faster than M. To be more quantitative, we may combine the hydrostatic equation... 

The barotropic flow in a channel of small width W and shallow depth H is treated as propagation of long waves in several text books of fluid dynamics, for example. Gill (1982). The pressure is assumed to satisfy the hydrostatic equation... 

Note also that in typical textbooks on mechanical engineering this relation will be given as T = —pe, as the hydrostatic equation yields ... 

The decrease of temperature with increasing altitude is called the lapse rate. Combining the hydrostatic equation and the expression for potential temperature, we can evaluate the dry adiabatic lapse rate (Td), i.e., the temperature decrease which would be associated with a vertical adiabatic displacement. Logarithmic differentiation of equation (3.9)... 

As can be seen from the hydrostatic equation (3.12), there exists a single-valued monotonic relationship between pressure and geometric height in a static atmosphere. Therefore, meteorologists often use the pressure (rather than the altitude) as the independent vertical coordinate. It is also convenient to use the log-pressure altitude... 

We have already derived the change of pressure with height in the atmosphere using the hydrostatic equation to derive (1.3). Substituting the dp/dz term and simplifying, we get... 

In fluid dynamics this is known as the equation of motion and applies to all types of fluids in motion or at rest. The only restriction is that gravity represents the external force field for hydrodynamics or hydrostatics. The hydrostatic equation (i.e., Vp = pg) is obtained by ignoring all terms that contain v and t. Each term has units of force per unit volume. 

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