Variational Data Assimilation for the Global Ocean - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 306

In the isopycnal option, observation or grid point differences in density are scaled

by ¡ s to form a correlation. This procedure essentially derives the vertical corre-

lations relative to a density vertical coordinate. Observations are more correlated
along an isopycnal than across an isopycnal, which introduces considerable flow
dependence into the correlations. The procedure is cost free and does not require a
transformation of the model background to isopycnal coordinates. All that is needed
is knowledge of the density for any point of interest, which can be obtained from
the observation itself or the model forecast. Use of the isopycnal vertical correlation
option is ideally suited for HYCOM, since each coordinate surface in the model is
assigned a reference isopycnal. Vertical correlation defined along isentropic surfaces
is well known in atmospheric data assimilation (e.g., Riishøjgaard 1998 ). Note
that vertical correlations in the analysis are calculated either via a SOAR, ( 13.4 )
or Gaussian, ( 13.5 ) function using lengths scales derived from either the vertical
density gradient or isopycnal formulations.
Figure 13.4 gives cross sections through the vertical correlation length scale field
and the model density field for the HYCOM Pacific domain (Sect. 13.6 ). The length
scales were computed using the vertical density gradient option with ¡ s D 0:15 .The

42 ı N, 140 ı E along a great circle cross sections extend from the coast of Japan at

160 ı E. Figure 13.4 a shows vertical correlation length
scales shorter near the surface and longer at depth in agreement with the density
stratification (Fig. 13.4 b). The influence of the Kuroshio front is clearly seen, with
longer length scales at increasingly shallower depths as the permanent thermocline
shoals towards the equator. Relatively longer length scales are also seen in the
17

0 ı N,

path to the equator at

19 ı C mode-water layer immediately south of the Kuroshio, which has relatively
uniform density at depths of 200-400 m.

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13.3.3 Multivariate Correlations

The horizontal and vertical correlation functions described above are used in the
analysis of temperature, salinity, and geopotential. Temperature and salinity are
analyzed as uncorrelated scalars, while the analysis of geopotential is multivariate
with velocity. Geopotential is computed in the analysis from vertical profiles of
temperature and salinity by integrating the specific volume anomaly ( Fofonoff
and Millard 1983 ) from a level of no motion (2,000 m depth) to the surface. The
multivariate correlations require specification of a parameter , which measures the

divergence permitted in the velocity correlations, and a parameter , which specifies

the strength of the geostrophic coupling of the velocity/geopotential correlations.
Typically,

'

) that produces weakly
divergent velocity increments and assumes that the divergence is not correlated
with changes in the mass field. The geostrophic coupling parameter

is set to a small, constant value ( D 0:05

' varies with

1 ı of latitude from the equator,
where geostrophy is not defined, and in shallow water (

location from 0 to 1. It is scaled to zero within

m deep), where friction
rather than pressure gradient forces control ocean flow. The multivariate correlations

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