Momentum flux at the air-sea interface

[Jesper Larsen]

Hi GETM users

We are looking into improving our sea level forecasts. We have already had a great deal of success on tuning the bathymetry. But now we also want to look at the formulation for air-sea momentum exchange.

Currently the wind speed is used in the formulations for the exchanges. At lower wind speeds neglect of currents can however lead to a systematic overestimation of the fluxes on the order of ~2-4% since winds and currents are to some degree covarying (but in the order of 20% for the Equatorial region), see Kara et al (2007). For a typical wind speed of 10 m/s (at 10 m) we might in some cases have a (systematic) surface current component along the wind speed direction in the order of 0.1 m/s. The formulation for the exchange coefficients assume that you use a relative wind speed, see Fairall et al (2003). This means that we are overestimating the momentum flux by:

W^2/(W-U)^2 ~ 2%

The same goes for latent and sensible heat fluxes. I am not to worried about an error of this size (even if it is systematic). But the fix for correcting it is trivial. What is your opinion on this? If we fix it we will check the effect on at least sea level in a one year hindcast.

Wave motion does also have an impact on the effective relative velocity of the wind and the sea surface (since waves are generally in the direction of the wind). But that is beyond our ambitions at this stage to address.

Another issue that we want to look at is the formulation of the calculation of the exchange coefficients. Is there a scientific reference for the current formulation for the calculation of the air-sea exchange coefficients (momentum, latent and sensible heat) in exchange_coefficients.F90?

As far as we can see the formulation is not quite up to date. Kara et al (2005) made a relatively new formulation based on a polynomial approximation to the COARE model (Fairall et al (2003)). We are considering updating the current algorithm to that algorithm (which is the one used in HYCOM). Any thoughts on that?

A fourth issue is that the bulk algorithms for air-sea exchanges are "atmospheric-centric" in the sense that they as far as I know do not take the boundary layer of the sea into account. We will also look into that.

We normally use the model values (of sea water temperature, currents, ...) for the first level in GETM. In the advanced model of Fairall et al (2003) there is correction of sea water temperature at some depth to the skin temperature. But it is not clear to me that this is the case in the approximations. This can potentially lead to an error in SST of some tenths of degrees in the formulation (which is in the order of magnitude of the model SST accuracy). It does also have an effect on the momentum flux. Does anyone know how to handle this? I guess it is possible to make an approximation based on the Fairall et al (2003) submodel for the ocean boundary layer?

Best regards
Jesper Baasch-Larsen

References:
Kara et al (2005): Stability-Dependent Exchange Coefficients for Air-Sea Fluxes. GRL v.34.
Kara et al (2007): Wave and current effects on the wind drag. J. Atmospheric and Oceanic Technology v.22.
Fairall et al (2003): Bulk Parameterization of Air-Sea Fluxes: Updates and Verification for the COARE Algorithm. J. Climate v.16.

[Jesper Larsen]

W^2/(W-U)^2 ~ 2%

should have been

W^2/(W-U)^2 ~ 1.02

[Ulf Gräwe]

On Fri, 2012-01-13 at 13:46 +0100, Jesper Larsen wrote:
> Hi GETM users
>
> We are looking into improving our sea level forecasts. We have already had a great deal of success on tuning the bathymetry. But now we also want to look at the formulation for air-sea momentum exchange.
>
> Currently the wind speed is used in the formulations for the exchanges. At lower wind speeds neglect of currents can however lead to a systematic overestimation of the fluxes on the order of ~2-4% since winds and currents are to some degree covarying (but in the order of 20% for the Equatorial region), see Kara et al (2007). For a typical wind speed of 10 m/s (at 10 m) we might in some cases have a (systematic) surface current component along the wind speed direction in the order of 0.1 m/s. The formulation for the exchange coefficients assume that you use a relative wind speed, see Fairall et al (2003). This means that we are overestimating the momentum flux by:
>
> W^2/(W-U)^2 ~ 2%

As far as I remember, we had such a discussion some months (years?) ago.
And I think that we agreed that we correct the wind speed by the
velocity of the uppermost layer. Am I right?
As an argument we used the periodic channel with constant wind. Without
correction the fluid will always be accelerated. Whereas in reality, if
the currents are as fast as the wind, not momentum transfer is possible.
This correction would also take into account the momentum transfer from
water to air (although negligible, but observed in flumes).

>
> The same goes for latent and sensible heat fluxes. I am not to worried about an error of this size (even if it is systematic). But the fix for correcting it is trivial. What is your opinion on this? If we fix it we will check the effect on at least sea level in a one year hindcast.
>
> Wave motion does also have an impact on the effective relative velocity of the wind and the sea surface (since waves are generally in the direction of the wind). But that is beyond our ambitions at this stage to address.
>
> Another issue that we want to look at is the formulation of the calculation of the exchange coefficients. Is there a scientific reference for the current formulation for the calculation of the air-sea exchange coefficients (momentum, latent and sensible heat) in exchange_coefficients.F90?

The current bulk formulae are based on : Kondo, Junsei: Air-sea bulk
transfer coefficients in diabatic conditions, Boundary-Layer Meteorology
9, volume 9, Springer Netherlands, 91–112, 1975

> As far as we can see the formulation is not quite up to date. Kara et al (2005) made a relatively new formulation based on a polynomial approximation to the COARE model (Fairall et al (2003)). We are considering updating the current algorithm to that algorithm (which is the one used in HYCOM). Any thoughts on that?
>

We also had that discussion. If we implement a new method in GETM thats
fine. Since in GOTM Kondo and Fairall are already coded, why not use
first the GOTM meteo module and call it in GETM (like turbulence). If
you like to code Kara et al, why not doing that in GOTM. This would
simplify the maintenance of the code, testing and also the GOTM user
will profit from that.

> A fourth issue is that the bulk algorithms for air-sea exchanges are "atmospheric-centric" in the sense that they as far as I know do not take the boundary layer of the sea into account. We will also look into that.
>
> We normally use the model values (of sea water temperature, currents, ...) for the first level in GETM. In the advanced model of Fairall et al (2003) there is correction of sea water temperature at some depth to the skin temperature. But it is not clear to me that this is the case in the approximations. This can potentially lead to an error in SST of some tenths of degrees in the formulation (which is in the order of magnitude of the model SST accuracy). It does also have an effect on the momentum flux. Does anyone know how to handle this? I guess it is possible to make an approximation based on the Fairall et al (2003) submodel for the ocean boundary layer?
>

Fairall take into account the correction for the skin layer. The
original algorithm took also into account the formation of a thin
surface layer during the course of the day. They did some vertical
integration from the point of reference (T-point) up to the surface.
Furthermore, these values were stored to be used in the next meteo-step.

> Best regards
> Jesper Baasch-Larsen
>

BTH: How is the ice model doing?

--
Ulf Gräwe

Leibniz Institute for Baltic Sea Research Warnemuende
Dept. for Physical Oceanography and Instrumentation
Seestrasse 15
D-18119 Rostock-Warnemuende
Germany

Tel. : +49 (381) 5197-358
Fax : +49 (381) 5197-440

[Jesper Larsen]

Hi Ulf

Thanks for your reply.

> On Fri, 2012-01-13 at 13:46 +0100, Jesper Larsen wrote:
> > Hi GETM users
> >
> > We are looking into improving our sea level forecasts. We have
> already had a great deal of success on tuning the bathymetry. But now
> we also want to look at the formulation for air-sea momentum exchange.
> >
> > Currently the wind speed is used in the formulations for the
> exchanges. At lower wind speeds neglect of currents can however lead to
> a systematic overestimation of the fluxes on the order of ~2-4% since
> winds and currents are to some degree covarying (but in the order of
> 20% for the Equatorial region), see Kara et al (2007). For a typical
> wind speed of 10 m/s (at 10 m) we might in some cases have a
> (systematic) surface current component along the wind speed direction
> in the order of 0.1 m/s. The formulation for the exchange coefficients
> assume that you use a relative wind speed, see Fairall et al (2003).
> This means that we are overestimating the momentum flux by:
> >
> > W^2/(W-U)^2 ~ 2%
>
> As far as I remember, we had such a discussion some months (years?)
> ago.
> And I think that we agreed that we correct the wind speed by the
> velocity of the uppermost layer. Am I right?

I do not believe that such a correction is performed (see meteo.F90).

> As an argument we used the periodic channel with constant wind. Without
> correction the fluid will always be accelerated. Whereas in reality, if
> the currents are as fast as the wind, not momentum transfer is
> possible.
> This correction would also take into account the momentum transfer from
> water to air (although negligible, but observed in flumes).

I agree on your argument. Although it does assume no waves but that is another discussion.

> > The same goes for latent and sensible heat fluxes. I am not to
> worried about an error of this size (even if it is systematic). But the
> fix for correcting it is trivial. What is your opinion on this? If we
> fix it we will check the effect on at least sea level in a one year
> hindcast.
> >
> > Wave motion does also have an impact on the effective relative
> velocity of the wind and the sea surface (since waves are generally in
> the direction of the wind). But that is beyond our ambitions at this
> stage to address.
> >
> > Another issue that we want to look at is the formulation of the
> calculation of the exchange coefficients. Is there a scientific
> reference for the current formulation for the calculation of the air-
> sea exchange coefficients (momentum, latent and sensible heat) in
> exchange_coefficients.F90?
>
>
> The current bulk formulae are based on : Kondo, Junsei: Air-sea bulk
> transfer coefficients in diabatic conditions, Boundary-Layer
> Meteorology
> 9, volume 9, Springer Netherlands, 91–112, 1975

Thanks.

> > As far as we can see the formulation is not quite up to date. Kara et
> al (2005) made a relatively new formulation based on a polynomial
> approximation to the COARE model (Fairall et al (2003)). We are
> considering updating the current algorithm to that algorithm (which is
> the one used in HYCOM). Any thoughts on that?
> >
>
> We also had that discussion. If we implement a new method in GETM thats
> fine. Since in GOTM Kondo and Fairall are already coded, why not use
> first the GOTM meteo module and call it in GETM (like turbulence). If
> you like to code Kara et al, why not doing that in GOTM. This would
> simplify the maintenance of the code, testing and also the GOTM user
> will profit from that.

Thanks for your suggestion. I will look more into this. The reason I suggested Kara et al instead of Fairall et al is performance. But maybe it is not a problem.

> > A fourth issue is that the bulk algorithms for air-sea exchanges are
> "atmospheric-centric" in the sense that they as far as I know do not
> take the boundary layer of the sea into account. We will also look into
> that.
> >
> > We normally use the model values (of sea water temperature, currents,
> ...) for the first level in GETM. In the advanced model of Fairall et
> al (2003) there is correction of sea water temperature at some depth to
> the skin temperature. But it is not clear to me that this is the case
> in the approximations. This can potentially lead to an error in SST of
> some tenths of degrees in the formulation (which is in the order of
> magnitude of the model SST accuracy). It does also have an effect on
> the momentum flux. Does anyone know how to handle this? I guess it is
> possible to make an approximation based on the Fairall et al (2003)
> submodel for the ocean boundary layer?
> >
>
> Fairall take into account the correction for the skin layer. The
> original algorithm took also into account the formation of a thin
> surface layer during the course of the day. They did some vertical
> integration from the point of reference (T-point) up to the surface.
> Furthermore, these values were stored to be used in the next meteo-
> step.

Yes, Fairall et al do take it into account. But I do not believe it is taken into account in the less advanced bulk formulations. But maybe we can do some tests on it using the GOTM Fairall et al flux coefficient formulation to see if it is important.

> BTH: How is the ice model doing?

Not too good unfortunately. Some of the problematic issues I encountered were quite far from my areas of expertise. And given other priorities we put it on hold. We are however considering getting outside expertise to help finish the work. But nothing has been decided on this issue yet.

> --
> Ulf Gräwe

- Jesper

[Adolf STIPS]

Dear Jesper,

definitely a good initiative, hopefully you will have more success then me by
updating the air-sea code (but what happened to the ice, did you stop this now?).

You probably do not know, but since December 2004 TOGA Coare air-sea routines
had been coded and tested by me. Due to decisions not under my control
these routines never made it into official GETM releases.
(Only after sending the code to Hans in 2007 it entered finally at least GOTM).
Besides this an improved albedo calculation for the solar radiation was also done.

Because of this personal not encouraging experience I'm not really optimistic
what will enter finally the GETM code.
But if there is any chance that it might enter I could volunteer to check what are doing, I have
test routines and test values for many situations.

Some comments below:

>
>
> Currently the wind speed is used in the formulations for the exchanges. At lower wind speeds neglect of currents can however lead to a systematic overestimation of the fluxes on the order of ~2-4% since winds and currents are to some degree covarying (but in the order of 20% for the Equatorial region), see Kara et al (2007). For a typical wind speed of 10 m/s (at 10 m) we might in some cases have a (systematic) surface current component along the wind speed direction in the order of 0.1 m/s. The formulation for the exchange coefficients assume that you use a relative wind speed, see Fairall et al (2003). This means that we are overestimating the momentum flux by:
>
> W^2/(W-U)^2 ~ 2%
>

Principally correct (but even more important for GOTM, where I had already proposed/submitted a patch, with no response).
This should not be a problem to implement, but I bet that the changes for the Baltic domain will be very small.
Instead of windspeed you use U10-Usurf, V10-Vsurf.

> The same goes for latent and sensible heat fluxes. I am not to worried about an error of this size (even if it is systematic). But the fix for correcting it is trivial. What is your opinion on this? If we fix it we will check the effect on at least sea level in a one year hind cast.

This is automatically corrected for because of the above.

> Wave motion does also have an impact on the effective relative velocity of the wind and the sea surface (since waves are generally in the direction of the wind). But that is beyond our ambitions at this stage to address.
>
> Another issue that we want to look at is the formulation of the calculation of the exchange coefficients. Is there a scientific reference for the current formulation for the calculation of the air-sea exchange coefficients (momentum, latent and sensible heat) in exchange_coefficients.F90?
>

Kondo 1975 - this is cited in the code!

> As far as we can see the formulation is not quite up to date. Kara et al (2005) made a relatively new formulation based on a polynomial approximation to the COARE model (Fairall et al (2003)). We are considering updating the current algorithm to that algorithm (which is the one used in HYCOM). Any thoughts on that?
>

I'm aware of that (and have coded that too), so we should go for that, as it is numerical more effective as the original TOGA COARE.

> A fourth issue is that the bulk algorithms for air-sea exchanges are "atmospheric-centric" in the sense that they as far as I know do not take the boundary layer of the sea into account. We will also look into that.
>
> We normally use the model values (of sea water temperature, currents, ...) for the first level in GETM. In the advanced model of Fairall et al (2003) there is correction of sea water temperature at some depth to the skin temperature. But it is not clear to me that this is the case in the approximations. This can potentially lead to an error in SST of some tenths of degrees in the formulation (which is in the order of magnitude of the model SST accuracy). It does also have an effect on the momentum flux. Does anyone know how to handle this? I guess it is possible to make an approximation based on the Fairall et al (2003) submodel for the ocean boundary layer?
>

This is another issue.
Skin temperature is NOT equal to SST.
It is not important for fluxes, as it is included in most parameterizations already.
It is/could be important if you compare to radiometric surface temperature measurements (from ship or satellite).
Using only night time scenes from satellite avoids any bias from skin to SST difference.

Cheers, Adolf

[Adolf STIPS]

Hi Ulf and Jesper,

We also had that discussion. If we implement a new method in GETM thats

fine. Since in GOTM Kondo and Fairall are already coded, why not use

first the GOTM meteo module and call it in GETM (like turbulence). If

you like to code Kara et al, why not doing that in GOTM. This would

simplify the maintenance of the code, testing and also the GOTM user

will profit from that.

Thanks for your suggestion. I will look more into this. The reason I suggested Kara et al instead of Fairall et al is performance. But maybe it is not a problem.

This points directly to one of the underlying issues why no progress had been made, as

one of the arguments was always to use only GOTM routines.

But for GETM really Kara would be more efficient, whereas for GOTM the original is okay 

and likely even more accurate (Kara is only an approximation of COARE).

For me this putting all to GOTM does not make sense, the only real solution is to have a separate air-sea module.

The GETM routines are more specific and more critical, why to bother about GOTM for that?

In order to avoid doubling routines no new routines were introduced for years!

BTH: How is the ice model doing?

Not too good unfortunately. Some of the problematic issues I encountered were quite far from my areas of expertise. And given other priorities we put it on hold. We are however considering getting outside expertise to help finish the work. But nothing has been decided on this issue yet.

The first point is bad, but may be in the longer term there is hope.

But did you not even try the very simple 0d ice model I've sent you?

(The patch had worked as e.g Gennadi has used it for the Baltic Sea)

Anyhow, saluti, Adolf

[Jesper Larsen]

Hi Adolf

> definitely a good initiative, hopefully you will have more success then
> me by
> updating the air-sea code (but what happened to the ice, did you stop
> this now?).

Please see my reply to Ulf.

> You probably do not know, but since December 2004 TOGA Coare air-sea
> routines
> had been coded and tested by me. Due to decisions not under my control
> these routines never made it into official GETM releases.
> (Only after sending the code to Hans in 2007 it entered finally at
> least GOTM).
> Besides this an improved albedo calculation for the solar radiation was
> also done.
>
> Because of this personal not encouraging experience I'm not really
> optimistic
> what will enter finally the GETM code.
> But if there is any chance that it might enter I could volunteer to
> check what are doing, I have
> test routines and test values for many situations.

Thanks for the offer.

> Some comments below:
> >
> > Currently the wind speed is used in the formulations for the
> exchanges. At lower wind speeds neglect of currents can however lead to
> a systematic overestimation of the fluxes on the order of ~2-4% since
> winds and currents are to some degree covarying (but in the order of
> 20% for the Equatorial region), see Kara et al (2007). For a typical
> wind speed of 10 m/s (at 10 m) we might in some cases have a
> (systematic) surface current component along the wind speed direction
> in the order of 0.1 m/s. The formulation for the exchange coefficients
> assume that you use a relative wind speed, see Fairall et al (2003).
> This means that we are overestimating the momentum flux by:
> >
> > W^2/(W-U)^2 ~ 2%
> >
>
> Principally correct (but even more important for GOTM, where I had
> already proposed/submitted a patch, with no response).
> This should not be a problem to implement, but I bet that the changes
> for the Baltic domain will be very small.

I believe you are right. But compared to the effort we have to put into it (which is very little) it might still be worth it. And our sea level forecasts (which are very important for us) are already very good. So even one or two percent extra in explained variance is a good result for us given the small amount of work it requires.

> Instead of windspeed you use U10-Usurf, V10-Vsurf.
>
> > The same goes for latent and sensible heat fluxes. I am not to
> worried about an error of this size (even if it is systematic). But the
> fix for correcting it is trivial. What is your opinion on this? If we
> fix it we will check the effect on at least sea level in a one year
> hind cast.
>
> This is automatically corrected for because of the above.
>
> > Wave motion does also have an impact on the effective relative
> velocity of the wind and the sea surface (since waves are generally in
> the direction of the wind). But that is beyond our ambitions at this
> stage to address.
> >
> > Another issue that we want to look at is the formulation of the
> calculation of the exchange coefficients. Is there a scientific
> reference for the current formulation for the calculation of the air-
> sea exchange coefficients (momentum, latent and sensible heat) in
> exchange_coefficients.F90?
> >
>
> Kondo 1975 - this is cited in the code!

I can't seem to find the reference in the code.

> > As far as we can see the formulation is not quite up to date. Kara et
> al (2005) made a relatively new formulation based on a polynomial
> approximation to the COARE model (Fairall et al (2003)). We are
> considering updating the current algorithm to that algorithm (which is
> the one used in HYCOM). Any thoughts on that?
> >
>
> I'm aware of that (and have coded that too), so we should go for that,
> as it is numerical more effective as the original TOGA COARE.

Is numerical efficiency important in this case? In the Kara et al (2005) paper they state that their algorithm takes 4 times less than the Fairall algorithm. But has anyone timed how big a fraction the air-sea fluxes take of the total CPU time?

> > A fourth issue is that the bulk algorithms for air-sea exchanges are
> "atmospheric-centric" in the sense that they as far as I know do not
> take the boundary layer of the sea into account. We will also look into
> that.
> >
> > We normally use the model values (of sea water temperature, currents,
> ...) for the first level in GETM. In the advanced model of Fairall et
> al (2003) there is correction of sea water temperature at some depth to
> the skin temperature. But it is not clear to me that this is the case
> in the approximations. This can potentially lead to an error in SST of
> some tenths of degrees in the formulation (which is in the order of
> magnitude of the model SST accuracy). It does also have an effect on
> the momentum flux. Does anyone know how to handle this? I guess it is
> possible to make an approximation based on the Fairall et al (2003)
> submodel for the ocean boundary layer?
> >
>
> This is another issue.
> Skin temperature is NOT equal to SST.
> It is not important for fluxes, as it is included in most
> parameterizations already.

Ok. But what depth have they used for the SST temperature? The temperature can vary significantly with depth (mainly during daytime). We have for example upper layer thicknesses ranging from maybe 5 cm to 40 cm in our North Sea / Baltic Sea setup. Others probably have much thicker layers. But the flux parameterizations assume that the "correction" from these temperatures to the skin temperature (which is what is important for the air-sea exchange) are identical (except Fairall which has a submodel for converting the SST at some depth to skin SST).

> It is/could be important if you compare to radiometric surface
> temperature measurements (from ship or satellite).
> Using only night time scenes from satellite avoids any bias from skin
> to SST difference.

Due to the "cool skin" effect there is even at nighttime a temperature difference in the order of 0.1 degC. But you can easily correct for this.

I believe that we have a similar problem for the flux formulations. Otherwise the fluxes would in some way or another have to depend on the upper layer thickness (or rather include a submodel/parameterization which takes into account that the ocean model upper layer temperatures represents different quantities at different places). I am however not sure if this effect is significant or not.

> Cheers, Adolf

Have a nice weekend

Jesper

[Adolf STIPS]

Hi Jesper,

>
>>> As far as we can see the formulation is not quite up to date. Kara et
>> al (2005) made a relatively new formulation based on a polynomial
>> approximation to the COARE model (Fairall et al (2003)). We are
>> considering updating the current algorithm to that algorithm (which is
>> the one used in HYCOM). Any thoughts on that?
>>>
>>
>> I'm aware of that (and have coded that too), so we should go for that,
>> as it is numerical more effective as the original TOGA COARE.
>
> Is numerical efficiency important in this case? In the Kara et al (2005) paper they state that their algorithm takes 4 times less than the Fairall algorithm. But has anyone timed how big a fraction the air-sea fluxes take of the total CPU time?
>

But you cannot compare to actual code (Kondo) and Kara is even more time consuming then Kondo.
There is no actual comparison done, but if I remember correct the difference
in runtime was considerable (~5%) when I did tests (but that was with simple IP methods).

>> This is another issue.
>> Skin temperature is NOT equal to SST.
>> It is not important for fluxes, as it is included in most
>> parameterizations already.
>
> Ok. But what depth have they used for the SST temperature? The temperature can vary significantly with depth (mainly during daytime). We have for example upper layer thicknesses ranging from maybe 5 cm to 40 cm in our North Sea / Baltic Sea setup. Others probably have much thicker layers. But the flux parameterizations assume that the "correction" from these temperatures to the skin temperature (which is what is important for the air-sea exchange) are identical (except Fairall which has a submodel for converting the SST at some depth to skin SST).
>

You right in the sense that all heat flux passes also the skin layer.

But this is irrelevant for GETM applications ( we do not compare to radiometric measurements do you?)

The ONLY reason why skin temperature is in TOGA COARE, is because they used
infrared radiometers for measuring the skin temperature.

Since 20 years many scientists (and in 1993 I was one of them) tried
to find corrections for estimating SST from skin temperature
(mainly for getting SST from satellite instead of skin temperature).
This still has not led anywhere and for practical reasons you use now just night time scenes for SST!

The story is different if you calibrate the satellite to skin temperature, but still
estimating the flux from satellite remains a challenge.

>> It is/could be important if you compare to radiometric surface
>> temperature measurements (from ship or satellite).
>> Using only night time scenes from satellite avoids any bias from skin
>> to SST difference.
>
> Due to the "cool skin" effect there is even at nighttime a temperature difference in the order of 0.1 degC. But you can easily correct for this.
>

I do not agree.

The skin effect depends from the heat flux. If you have heat flux out of the water you do not have any cool skin,
which is the typical night (convective) situation.
Of course there are few situations when this is not the case,..

> I believe that we have a similar problem for the flux formulations. Otherwise the fluxes would in some way or another have to depend on the upper layer thickness (or rather include a submodel/parameterization which takes into account that the ocean model upper layer temperatures represents different quantities at different places). I am however not sure if this effect is significant or not.
>
>>

I do not understand this.
For simple bulk models the total heat content of the bulk layer is of interest, but in GETM we have heat flux
(also between layers and between grid points).

I'm quite sure that for all normal GETM applications we can safely forget the skin effect, this is not our problem.

Ciao, Adolf