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2.4 Vertical interpolations and extrapolations.

2.4.1 General considerations.

For 3D variables to be vertically interpolated, vertical interpolations are generally linear interpolations between the layers where are defined model variables. The treatment of the extrapolations above the upper layer, the extrapolations below the lower layer or the surface depend on the variable considered. In particular cases some variables can be diagnosed using the vertically interpolated value of some other variables.

The `OLD' versions of routines when available are used at ECMWF (switch LOLDPP=.T.). METEO-FRANCE uses the option LOLDPP=.F. .

2.4.2 More details for 3D dynamical variables.

Wind components, wind velocity.

Way of interpolating if LOLDPP=.T. (routine PPUV_OLD):

• Linear interpolation between the layer 2 and the lower layer.
• The coordinate used for linear interpolation is the logarithm of the pressure.
• Quadratic interpolation between the layer 1 and the layer 2 using the values of the layers 1, 2 and 3.
• Quadratic interpolation between the top and the layer 1 using the values of the top, layers 1 and 2; the value of the top is obtained by a linear extrapolation from the values of the layers 1 and 2.
• The coordinate used for quadratic interpolation is the logarithm of the pressure.
• Extrapolation below the middle of the lower layer and below the surface assumes that the quantity is constant.

Way of interpolating if LOLDPP=.F. (routine PPUV):

• The same as for LOLDPP=.T. but quadratic interpolations are replaced by linear interpolations.

Temperature.

Applies to temperature if the vertical coordinate of post-processing is not the potential vorticity, otherwise see routine PP2DINT.

Way of interpolating if LOLDPP=.F. (routine PPT):

• Quadratic interpolation between the middles of the upper and lower layers.
• Quadratic interpolation between the top and the middle of the upper layer: the top value of the temperature is assumed to be equal to the value of the middle of the upper layer; due to the fact that the interpolation is a quadratic one, that does not mean that the temperature is constant in this atmosphere depth.
• The coordinate used for quadratic interpolation is the logarithm of pressure. For more details about the quadratic interpolation used, which is a quadratic analytic expression of the logarithm of pressure, and the reason of using a quadratic interpolation, see (Undén, 1995).
• A surface temperature \TSURF is computed as follows:

(2.1)

• Extrapolation below the middle of the lower layer and the surface is a linear interpolation between and .
• Extrapolation under the surface is made according a more complicated algorithm:

(2.2)

• where:

(2.3)

• If ; if the expression for is more complicated:

(2.4)

• if :

(2.5)

• and : is computed by a linear interpolation (coordinate of interpolation is ) between the two values and

Way of interpolating if LOLDPP=.T. (routine PPT_OLD):

• Linear interpolation (between the upper and the lower layer).
• The coordinate used for linear interpolation is the pressure.
• Extrapolation above the middle of the upper layer assumes that the quantity is constant.
• Extrapolation below the middle of the lower layer and the surface is a linear interpolation between TL and \TSURF like in PPT.
• Extrapolation under the surface is made according the same algorithm as in PPT (code of part 1.4 is different in PPT and in PPT_OLD but actually does the same calculations).

Geopotential.

Applies to geopotential if the vertical coordinate of post-processing is not the potential vorticity, otherwise see routine PP2DINT.

Way of interpolating if LOLDPP=.T. (routine PPGEOP_OLD):

• The variable interpolated is a geopotential departure from a reference defined by a standard atmosphere without any orography. After the interpolation an increment is added, sum of the surface orography and the `standard' geopotential depth between the pressure level of interpolation and the actual surface. This method avoids to introduce interpolations for the standard component of the geopotential which can be computed analytically (in routine PPSTA).
• Linear interpolation between the layer 2 and the surface.
• The coordinate used for linear interpolation is the logarithm of the pressure.
• Quadratic interpolation between the layer 1 and the layer 2 using the values of the layers 1, 2 and 3.
• Quadratic interpolation between the top and the layer 1 using the values of the top, layers 1 and 2.
• The coordinate used for quadratic interpolation is the logarithm of the pressure.
• Extrapolation below surface uses the surface temperatur of Eq. (2.1).

(2.6)

• where is defined by formula (2.3) with in all cases.
• For more details about this algorithm, see (Andersson and Courtier, 1992) which is still valid for the cycles 17 and 18.

Way of interpolating if LOLDPP=.F. (routine PPGEOP):

• The variable interpolated is a geopotential departure from a reference defined by a standard atmosphere without any orography. After the interpolation an increment is added, sum of the surface orography and the "standard" geopotential depth between the pressure level of interpolation and the actual surface. This method avoids to introduce interpolations for the standard component of the geopotential which can be computed analytically (in routine PPSTA).
• Quadratic interpolation between the middles of the upper and lower layers.
• Quadratic interpolation between the top and the middle of the upper layer.
• The coordinate used for quadratic interpolation is the logarithm of pressure. The quadratic interpolation is not exactly the same as for LOLDPP=.T., it is a quadratic analytic expression of the logarithm of pressure of the same type as the one used to post-process the temperature for LOLDPP=.F. . For more details about the quadratic interpolation used, and the reason of using a quadratic interpolation, see (Undén, 1995).
• Linear interpolation between the lower layer and the surface, as for LOLDPP=.T. .
• Extrapolation below surface uses the same algorithm as for LOLDPP=.T. .

2.4.2 (a) Variables interpolated using routine PP2DINT.

List of variables:

• Geopotential if vertical coordinate is potential vorticity.
• Temperature if vertical coordinate is potential vorticity.
• Relative vorticity .
• Divergence .
• Potential temperature if vertical coordinate is not potential temperature.
• Velocity potential .
• Stream function .
• Equivalent potential temperature .
• Absolute vorticity .
• Stretching deformation .
• Shearing deformation .
• Potential vorticity .
• Wet potential vorticity (not yet coded).

Way of interpolating:

• Linear interpolation (between the upper and the lower layer for quantities defined on the middle of layers, between the layer 1 and the surface for quantities defined on interlayers).
• The coordinate used for linear interpolation is the pressure.
• Extrapolation above the middle of the upper layer assumes that the quantity is constant.
• Extrapolation below the middle of the lower layer and below the surface assumes that the quantity is constant.

Moisture, liquid water, solid water, cloud fraction, passive scalars: variables using routine PPQ.

Way of interpolating (routine PPQ):

• Linear interpolation (between the upper and the lower layer).
• The coordinate used for linear interpolation is the pressure.
• Extrapolation above the middle of the upper layer assumes that the quantity is constant.
• Extrapolation below the middle of the lower layer and below the surface assumes that the quantity is constant.

Relative humidity (routine PPRH).

Way of interpolating (routine PPRH):

• Linear interpolation (between the upper and the lower layer).
• The coordinate used for linear interpolation is the pressure.
• Extrapolation above the middle of the upper layer assumes that the quantity is constant.
• Extrapolation below the middle of the lower layer and below the surface assumes that the quantity is constant.

Pressure coordinate vertical velocity (routine PPVVEL).

Way of interpolating (routine PPVVEL):

• Linear interpolation (between the upper and the lower layer).
• The coordinate used for linear interpolation is the pressure.
• Extrapolation above the middle of the upper layer is a linear interpolation between a zero value at the top and the value of the upper layer.
• Extrapolation between the middle of the lower layer and the surface assumes that the quantity is constant.
• Extrapolation below the surface assumes that the quantity is zero.

Moist (irreversible) pseudo-adiabatic potential temperature (routine PPTHPW.

Routine PPTHPW is a diagnostic one. It takes as input the vertically post-processed pressure, temperature, moisture, liquid water and ice and computes at the post-processing levels using a diagnostic (and rather complicated) algorithm.

2.4.3 2D dynamical variables which need extrapolations.

Mean sea level pressure (routine PPPMER).

If is lower than 0.001 the mean sea level pressure is set to the surface pressure. In the other cases one uses the following algorithm:

• One computes the surface temperature of Eq. (2.1) and the `mean sea level' temperature

.

• To avoid extrapolation of too low pressures over high and warm surfaces the following modifications are done:
• if and , is defined by:

(2.7)

• if and , is set to 0, is modified and set to .
• To avoid extrapolation of too high pressures over cold surfaces the following modifications are done when : is set to and is modified and set to .
• In the other cases is set to .
• Mean sea level pressure is computed as follows:

(2.8)

• where:

(2.9)