Hydrostatic approximation

The pressure difference promoting vertical vent flows are solely due to temperature stratification, and conform to a hydrostatic approximation. In other words, the momentum equation in the vertical direction is essentially, due to the low flows ... 

Circulation models are based on the equations of motion of the geophysical fluid dynamics and on the thermodynamics of seawater. The model area is divided into finite size grid cells. The state of the ocean is described by the velocity, temperature, and salinity in each grid cell, and its time evolution can be computed from the three-dimensional model equations. To reduce the computational demands, the model ocean is usually incompressible and the vertical acceleration is neglected, the latter assumption is known as hydrostatic approximation. This removes sound waves in the ocean from the model solution. In the horizontal equations, the Boussinesq approximation is applied and small density changes are ignored except in the horizontal pressure gradient terms. This implies that such models conserve... 

Alternatively, the vertical grid cell size may vary horizontally following the depth variations. This is the so-called ct-coordinate approach, which is expected to give a better representation of dense inflowing water masses over varying bottom topography. However, the accurate calculation of baroclinic pressure gradients creates problems for this type of coordinates. Moreover, the vertical acceleration is also excluded by the hydrostatic approximation. 

The latter process also has to be parameterized because the model equations are based on the hydrostatic approximation. Convection is of special importance in the surface layer during the autumn and winter, and it is responsible for the formation of winter water layers in the deep basins of the Baltic Sea. Surface salinity well below 24 implies a maximum density at temperatures above the freezing point and a stable re-stratification before the freezing point is reached. Hence, a good approximation for the equation of state is required to simulate such processes, even if the existence of large density gradients could suggest that a simple and numerically cheap approximation could be sufficient. 

This simplification is commonly called the hydrostatic approximation. The approximation states that, in the vertical direction, the most important forces acting on a parcel of air are the vertical pressure gradient and gravity. This approximation is common in dynamical models, although it neglects some phenomena such as sound waves, and is not appropriate for high resolution models. The ideal gas law can be used to substitute for p in the above equation to obtain... 

These equations show that stratospheric winds blow approximately parallel to the contours of the geopotential field. An alternative version of the geostrophic balance can be obtained by substituting the hydrostatic approximation as given in (3.23c) into (3.36a,b), after differentiating with respect to z ... 

Hydrostatic approximation Approximation to the vertical momentum equation that balances the vertical component of the pressure gradient force, which is directed upward, with the downward gravitational force. 

For mai large-scale phenomena (e.g., midlatitude synoptic storms), the vertical pressure gradierrt is observed to be in very good balance with gravity. This balance yields the hydrostatic approximation. It is rigorously justified when the aspect ratio 8 = D/L < l (i.e., when the horizontal scale of the phenomena is significarrtly larger than the vertical scale). From Eq. (29c) the hydrostatic approximation is... 

Equations (60) and (61) constitute the quasi-geostrophic system. The rigorous justifications from scale analysis require that E 1 (so that friction may be neglected), 5 1 (so that the hydrostatic approximation is valid), and e 1 (so that the beta-plane approximation is valid). Furthermore, it is required that Ro < Cl. If each variable is expanded in terms of this small parameter, the balance of all terms of magnitude 1 in the resulting equations yields the geostrophic approximation. To obtain equations that accoimt for the time evolution, the balance of terms of size Ro must be considered. This balance yields the quasi-geostrophic system. 

The tracer 5 is zero in the hydrostatic approximation and unity otherwise. Here V is the horizontal gradierrt operator. These can be combined to obtain a single equation in... 

The vector F is the quasi-geostrophic EP flux. Its components are the zonally averaged northward eddy flux of zonal momentum and the zonally averaged northward eddy flux of temperature (because d dp is proportional to temperature through the hydrostatic approximation). Equation (117) shows that the EP flux divergence, defined by Eq. (115), induces changes in the basic state. 

The hydrostatic approximation permits definition of a new coordinate system, frequently used in dynamical meteorology, and one that is advantageous for our purposes here. The geometric height, z, is replaced by a new height variable... 

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