Large near-inertial oscillations of the Atlantic meridional overturning circulation

Ocean Modelling 42 (2012) 50–56

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Large near-inertial oscillations of the Atlantic meridional overturning circulation q Adam T. Blaker a,⇑, Joël J-M. Hirschi a, Bablu Sinha a, Beverly de Cuevas a, Steven Alderson a, Andrew Coward a, Gurvan Madec a,b a b

National Oceanography Centre, Southampton, Southampton, SO14 3ZH, UK LOCEAN (CNRS/IRD/UPMC/MNHN) Institut Pierre et Simon Laplace, Paris, France

a r t i c l e

i n f o

Article history: Received 17 August 2011 Received in revised form 15 November 2011 Accepted 18 November 2011 Available online 6 December 2011 Keywords: Meridional overturning circulation High frequency variability Near inertial gravity waves Ocean model

a b s t r a c t The Atlantic meridional overturning circulation (AMOC) is a key contributor to Europe’s mild climate. Both observations and models suggest that the AMOC strength varies on a wide range of timescales. Here we show the existence of previously unreported large near inertial AMOC oscillations in a high resolution ocean model. Peak-to-peak these oscillations can exceed 50 Sv (50  106 m3 s1) in one day. The AMOC oscillations are caused by equatorward propagating near-inertial gravity waves (NIGWs) which are forced by temporally changing wind forcing. The existence of NIGWs in the ocean is supported by observations, and a significant fraction of the ocean’s kinetic energy is associated with the near inertial frequencies. Our results also suggest that the NIGW-driven MOC variability would be near invisible to contemporary AMOC observing systems such as the RAPID MOC system at 26.5°N. Crown Copyright Ó 2011 Published by Elsevier Ltd. All rights reserved.

1. Introduction The Atlantic meridional overturning circulation (AMOC) is part of the global ocean conveyor which transports warm and saline surface waters to the North Atlantic (Broeker, 1987; Dickson and Brown, 1994; Kuhlbrodt et al., 2007). On their journey towards the Nordic seas these surface waters gradually become denser as they release heat to the atmosphere. Eventually, the increasing density leads to the sinking of the water masses and they are returned southward as cold and dense North Atlantic deep water. In the subtropical North Atlantic the surface and deep branches of the AMOC result in a maximum net northward heat transport of more than 1 PW (Ganachaud and Wunsch, 2000; Lumpkin and Speer, 2007; Johns et al., 2011). The AMOC has been identified as a key ocean mechanism which contributes to the comparatively mild European climate. A large fraction of the heat released to the atmosphere by the AMOC is carried eastward towards Europe by the predominant westerly winds, leading to warmer temperatures in northwestern Europe than at similar latitudes in western Canada (Rhines and Häkkinen, 2003; Broeker, 1987). Paleoclimate records suggest that in the past the AMOC is likely to have undergone major rapid fluctuations (McManus et al., 2004). Furthermore, most climate models suggest that under anthropogenic greenhouse gas forcing the AMOC will likely undergo a 30% q National Oceanography Centre, Southampton, University of Southampton Waterfront Campus, European Way, Southampton, SO14 3ZH, UK. ⇑ Corresponding author. Tel.: +44 (0)2380 596 312. E-mail address: [email protected] (A.T. Blaker).

reduction by the year 2100 (IPCC, 2007; Gregory et al., 2005). Any change in the strength of the AMOC would affect the northward transport of heat and therefore the climate over the North Atlantic region. Additionally, a change in the strength of the AMOC may also impact the capacity of the North Atlantic to absorb greenhouse gases. Recent observation-based studies have shown the AMOC transport at 26.5°N to exhibit substantial variability on short timescales (Cunningham et al., 2007; Kanzow et al., 2009; Kanzow et al., 2010). Between April 2004 and April 2009, the maximum AMOC transport at 26.5°N has a mean of 18.54 Sv, with a standard deviation of 4.68 Sv. The intra-annual peak-to-peak range of the AMOC can be as large as 30 Sv (Cunningham et al., 2007). The origin of the observed large AMOC variability is only partly understood. Some variability, such as the observed seasonality of the AMOC, can be linked to the seasonal variability in the wind stress curl along the African coast (Kanzow et al., 2010). The contribution to the observed AMOC variability by mechanisms such as baroclinic Kelvin waves (Kawase, 1987; Johnson and Marshall, 2002) or Rossby waves (Hirschi et al., 2007) has yet to be established. However, the existence of large AMOC variability on short timescales has been confirmed in both observations (Cunningham et al., 2007; Kanzow et al., 2009; Kanzow et al., 2010) and numerical ocean and coupled-climate models (Hirschi and Marotzke, 2007; Baehr et al., 2007; Baehr et al., 2009; Balan Sarojini et al., 2011). To better understand the AMOC variability on short timescales we use a global ocean/sea-ice model with a nominal grid resolution of 0.25°, sufficient to resolve western boundary current structures and to simulate the main processes associated with deep water formation necessary to model the AMOC. Two model integrations

1463-5003/$ - see front matter Crown Copyright Ó 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ocemod.2011.11.008

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A.T. Blaker et al. / Ocean Modelling 42 (2012) 50–56

where model output is stored at high frequency reveal some interesting large amplitude variability in the AMOC. 2. Description of the model and numerical integrations We use NEMO v3.0 (Nucleus for European Modelling of the Ocean) (Madec, 2008) in the Global ORCA025 configuration set up in the DRAKKAR project (DRAKKAR Group, 2007; Barnier et al., 2006; Madec, 2008). Horizontal resolution is 1/4° (1442  1021 grid points). South of 20°N the model grid is isotropic Mercator, and north of 20°N the grid becomes quasi-isotropic bipolar, with poles located in Canada and Siberia to avoid the numerical instability associated with the convergence of the meridians at the geographic north pole. At the Equator the resolution is approximately 27.75 km, becoming finer at higher latitudes such that at 60°N/S it becomes 13.8 km. The model has 64 vertical levels with a grid spacing increasing from 6 m near the surface to 250 m at 5500 m. Bottom topography is represented as partial steps and bathymetry is derived from ETOPO2 (U.S. Department of Commerce, 2006). To prevent excessive drifts in global salinity due to deficiencies in the fresh water forcing, sea surface salinity is relaxed towards climatology with a piston velocity of 33.33 mm/day/psu. Sea ice is represented by the Louvain-la-Neuve Ice Model version 2 (LIM2) sea-ice model (Timmerman et al., 2005). Starting from rest the model simulates the period 1958–2001, with surface forcing comprising of 6-hourly mean momentum fields, daily mean radiation fields and monthly mean precipitation fields supplied by the DFS4.1 dataset (Brodeau et al., 2010) and linearly interpolated from the time mean fields by the model. Model output is stored as 5-day averages. Starting from the ocean state on January 1st 1989 in the simulation described above, we perform two additional NEMO ORCA025 integrations where a 2-month period (January 1st 1989 to February 28th 1989) is re-run and where model output is stored as 4-hourly averages. The first integration uses the same surface forcing as before. In the second integration the wind forcing is held temporally constant by applying the wind forcing on the first model timestep to all subsequent integration timesteps.

Fig. 1. Latitude-depth plots of the AMOC (Sv) for the period 1989–1995: (a), time mean and (b), standard deviation based on 5 day mean model output.

Wmod ¼ Wbtr þ Wgeo þ Wekm þ Wres

Wbtr is the barotropic component arising from depth averaged velocities,

Wbtr ðy; zÞ ¼

Wmod ðy; zÞ ¼

z

Hmax ðyÞ

xe

v ðx; y; z0 Þ dxdz0 ;

ð1Þ

xw

where xw and xe are the western and eastern boundaries of the ocean basin respectively, Hmax(y) is the maximum depth of the ocean basin and z is the vertical coordinate. The time mean AMOC in NEMO (Fig. 1) compares well with other models. Typical of most ocean models, the return flow in the deep ocean is too shallow, by a few hundred metres, but the depth of the maximum value, at 1000 m, compares well with observations (Kanzow et al., 2009). At 26.5°N and 1000 m depth the AMOC calculated from a typical 5 year period, equivalent to the current length of the RAPID 26.5°N observations, is 22.54 Sv, 4 Sv higher than RAPID observations, and the standard deviation is 4.1 Sv, 10% lower than observed.

xe

v ðx; yÞ dxdz0

ð3Þ

v ðx; y; z0 Þ dz0

ð4Þ

xw

in which

v ðx; yÞ ¼

1 Hðx; yÞ

Z

0

Hðx;yÞ

is the barotropic meridional velocity. Wgeo(y, z), the baroclinic geostrophic (or thermal wind) component arising from zonal density gradients across the Atlantic basin is

Z

Wgeo ðy; zÞ ¼ where

Z

Z

z

Hmax ðyÞ

3. Analysis and decomposition of the AMOC

Z

Z

Z

z

xe

0

ðv geo ðx; y; z0 Þ  v geo ðx; yÞÞ dxdz ;

ð5Þ

xw

Hmax ðyÞ

We calculate the AMOC, Wmod(y, z), as the zonal and vertical integral of v (x, y, z), the meridional velocity, i.e.

ð2Þ

vgeo and v geo are

v geo ðx; y; zÞ ¼ 

g

Z

q

f

z

@q 0 dz @x

Hðx;yÞ

ð6Þ

and

v geo ðx; yÞ ¼

1 Hðx; yÞ

Z

0

v geo ðx; z0 Þ dz0

ð7Þ

Hðx;yÞ

respectively, g being the Earth’s gravitational acceleration, q the in situ density, f the Coriolis parameter, and q⁄ a reference density. Wekm(y, z), the Ekman (wind driven) component compensated by a section mean return flow to ensure no net transport is,

Wekm ðy; zÞ ¼

Z

Z

z

Hmax ðyÞ

where

xe

0

ðv ekm ðx; y; z0 Þ  v ekm ðx; yÞÞ dxdz ;

ð8Þ

xw

vekm(x, y, z0 ) and v ekm ðx; yÞ are

v ekm ðx; y; zÞ ¼ 

3.1. Decomposition of the AMOC

1 ðq fLDz Þ

Z

xe

sx dx

ð9Þ

xw

and The model AMOC, Wmod, can be decomposed into one barotropic and three baroclinic component parts, each representing a different physical process and corresponding to the components which are measured by the RAPID observing array (Hirschi and Marotzke, 2007),

v ekm ðx; yÞ ¼ 

1 ðq fAÞ

Z

xe

sx dx

ð10Þ

xw

respectively, L being the basin width, Dz the Ekman depth, and A the cross-sectional area of the basin. The Ekman depth, Dz, which

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A.T. Blaker et al. / Ocean Modelling 42 (2012) 50–56

defines the base of the Ekman layer in which the wind driven transport occurs is typically chosen to be around 50–100 m. The choice does not strongly affect the resulting overturning profile. Wres is the baroclinic ageostrophic residual obtained by rearranging Eq. (2),

Wres ¼ Wmod  ðWbtr þ Wgeo þ Wekm Þ:

ð11Þ

3.2. Analysis of the AMOC decomposition From the decomposition of the AMOC at 26.5°N, using the 5-day mean output, it is seen that the majority of the variability comes from the Wbtr and Wekm components (Fig. 2a). The Wgeo term also shows variability, but at a lower frequency, and the Wres term remains close to zero with little variability throughout the integration. However, the 4-hourly mean output (Fig. 2b) reveals some interesting details. The AMOC exhibits strong variability with a period close to 1 day. The daily range of 4-hourly mean AMOC values exceeds 50 Sv on several occasions (e.g. around January 5th). This is more than the maximum AMOC range seen for 5-day averages for the 1958 to 2001 period. The decomposition of the 4-hourly mean AMOC timeseries provides some insight into the origin of the large daily variability. Whilst for 5-day means Wres shows almost no variability, this component dominates the 4-hourly mean timeseries. However, the higher temporal resolution does not reveal similar increases in the variability of Wbtr, Wekm and Wgeo. The standard deviation of the AMOC variability calculated from the 4-hourly mean model output is much larger (8–12 Sv, Fig. 3a) than that calculated from the 5-day mean model output (3–5 Sv, Fig. 1). Fig. 3 also shows that the increased AMOC variability at high temporal resolution occurs at all latitudes with the largest amplitudes found between 25°N and 45°N and south of 15°N. The standard deviation for Wres (Fig. 3b) shows that 5–8 Sv are attributable to this component. South of 35°N the maximum variability of Wres is found between 1500 and 2500 m depth, which is substantially deeper than the maximum AMOC depth seen for 5day averages (see Fig. 1). North of 35°N the depth of maximum variability of Wres decreases to depths of 1000 m or less. The standard deviation for Wmod  Wres (Fig. 3c), shows that the other components typically account for 3–6 Sv, with a peak of 8 Sv between 38°N and 48°N. The maximum variability for Wmod  Wres occurs

a

b

Fig. 3. Standard deviation of the AMOC (Sv) for (a) the two months of 4-hourly mean data, (b) for the residual component of the AMOC and (c) the standard deviation for the 4-hourly AMOC data with the residual component subtracted.

in the top few hundred metres. This confinement to the surface reflects variability in Wek as both wind strength and variability reach a maximum during the boreal winter months. A useful insight into the spatio-temporal behaviour of the AMOC is obtained by looking at the AMOC at 1000 m depth as a function of latitude and time (Fig. 4a). The AMOC is characterised

40 30 20 10 0 -10 -20

60 40 20 0 -20

Fig. 2. Time series of the AMOC, Wmod, the barotropic, Wbtr, geostrophic, Wgeo Ekman, Wekm, and residual, Wres, components (Sv) at 26.5°N for (a) the period 1985–1996 based on 5-day mean velocities and (b) a zoom showing the AMOC and its components for the period January–February 1989 based on 4-hourly mean velocities.

A.T. Blaker et al. / Ocean Modelling 42 (2012) 50–56

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by a striking series of peaks and troughs that propagate equatorwards. The propagation is particularly clear south of about 40°N where AMOC peaks (troughs) typically propagate from 40°N to 10°N in 7–10 days. As expected from the results described above, the wave-like propagation of AMOC anomalies is due to the variability found in Wres (Fig. 4b). No propagating features of this kind are found for the other AMOC components (not shown). An important clue about the nature of the propagating AMOC features is provided by the wave period and propagation speed which exhibit a latitudinal dependence with shorter period, faster waves at high latitudes, and propagation speeds becoming gradually slower and periods becoming longer towards the Equator. This behaviour is typical for near-inertial gravity waves (NIGWs) (Anderson and Gill, 1979). NIGWs are known to exist in the real ocean (Alford, 2003) and can be generated in idealised and realistic numerical models (Fox et al., 2000; Klein et al., 2004; Komori et al., 2008; Furuichi et al., 2008). However, a link between such waves and the AMOC has never been described before. 4. Near inertial gravity waves Having identified an equatorward propagating signal in the AMOC with properties reminiscent of NIGWs, we next examine the velocity fields output by the model. Fig. 5a shows the vertical velocity at 2 km depth for a typical 4-h mean taken from late January 1989. The domain is dominated by a series of internal waves with displacements as large as +/60 m per day and a characteristic wavelength of 150–200 km, slightly longer where the amplitudes are largest in the western half of the basin, to the south of the Gulf Stream. The Gulf Stream region exhibits a more broken field of strong positive and negative values, whilst for most of the rest of the basin there are regular wave crests and troughs aligned WSW-ENE, with individual wave crests (troughs) remaining well defined across a large fraction of the ocean basin. The waves are also highly coherent with depth (Fig. 5b), with maximum values located at mid-depth, although the waves can clearly be seen extending over the whole ocean depth. This suggests that

Fig. 4. Latitude-time plots of (a) the AMOC (Sv) and (b) the residual component Wres (Sv) at 1 km depth for the months January and February of 1989 for the experiment run with variable wind forcing.

Fig. 5. Vertical velocity (m/day) for a single 4 h mean from the control integration at (a) 2 km depth and (b) along 55°W (m/day). c shows the period of the peak power in the vertical velocity field (blue), and in the AMOC (red). The inertial period is also shown (black).

the waves are predominantly first baroclinic mode. Indication of higher modes (visible as alternating positive and negative vertical velocities in the water column) is found between 35°N and 40°N, in and immediately south of the generation area. The waves are most clearly visible in the vertical velocity field, but their imprint can also be seen in the horizontal velocity components, as well as in the sea surface height. In the surface layers strong zonal and meridional gradients (currents, mesoscale eddies) characterise meridional velocities and sea surface height so that the NIGWs do not stand out as clearly. As a second order effect NIGWs can also be seen in the temperature and salinity fields due to vertical displacement of the isopycnals. Computing the power spectrum of the wave field shown in Fig. 5a and b, and also of the residual component of the AMOC (Fig. 4b) reveals that the peak energy for both the wave field and the AMOC is at near inertial periods (Fig. 5c). From 35°N–65°N the wave period is inertial. Equatorward of 35°N the period is slightly shorter than the inertial period. We, like Komori et al. (2008), also see a peak in the power spectrum near 2f, but this is always smaller than the peak at f, so does not contribute to the analysis shown in Fig. 5c. Danioux et al. (2011) conduct idealised experiments using two different horizontal resolutions. They also find a peak at 2f, and interestingly they find that whilst it is smaller than the peak at f in the lower resolution, it is dominant in the experiments with the higher resolution.

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A.T. Blaker et al. / Ocean Modelling 42 (2012) 50–56

As mentioned earlier, NIGW fields of this type have been observed previously in other high resolution ocean and coupled ocean–atmosphere models (Fox et al., 2000; Komori et al., 2008), and have been investigated in idealised model configurations (Klein et al., 2004). After formation the waves propagate equatorward with a slight eastward component according to the beta- dispersion relation (Anderson and Gill, 1979), where propagation speed is fast at high latitudes and reduces with latitude. NIGWs in the ocean are mainly caused by wind variability, for example during the passage of winter storms at midlatitudes or of tropical cyclones (Thomson and Huggett, 1981). Conducting a second model experiment with 4-hourly mean output, identical to the first but with the wind forcing held constant, allows us to assess the impact of the wind variability on the near inertial AMOC oscillations. Without wind variability the near inertial AMOC variability vanishes within a few weeks (Fig. 6a and b). This demonstrates that the near inertial AMOC variability is forced by variability in the wind field. At mid- to high- latitudes, where propagation speeds are fastest, the waves are substantially reduced in amplitude after 14 days, and by 28 days there is little evidence of them remaining in the AMOC or the residual component. We also note that the wavelength and phase speed of the vanishing NIGWs both gradually decrease with time while their frequency remains essentially unchanged. This is because long NIGWs propagate faster and therefore leave the domain more quickly than their short counterparts. This is the expected dispersion for Poincaré waves (the family of waves to which NIGWs belong (Paldor et al., 2007; Glazman and Cheng, 1999)), and the similarity in the behaviour of the attenuation of the large high frequency AMOC oscillations and the NIGWs is another indication that they are a basin scale manifestation of NIGWs. 5. Discussion and conclusion NIGWs have been previously documented in high resolution ocean and coupled ocean–atmosphere models (Fox et al., 2000;

Fig. 6. Latitude-time plots of (a) the AMOC (Sv) and (b) the residual component Wres (Sv) at 1 km depth for the months January and February of 1989 for the experiment run with constant wind forcing.

Komori et al., 2008), but their influence on the AMOC has not. We have shown here that NIGWs exist in the NEMO ocean model, and that they excite variability in the AMOC on near inertial time scales far larger than the variability seen at longer timescales. On a daily timescale the range of near inertial AMOC variability can exceed 50 Sv. The NIGWs are driven by variability in the wind forcing fields and we show that they disappear if the wind is held constant. The wind forcing provided in DFS4.1 is derived from ERA40 (Uppala et al., 2005), with adjustments made using QuikSCAT (Liu et al., 1998) data to correct periods of anomalously low wind speeds seen in the ERA40 data. Horizontal resolution of the reanalysed wind field is 1.125°  1.125°, much coarser than the 0.25° ocean grid. A higher spatial and temporal resolution wind forcing, or indeed coupled atmosphere, would provide greater wind variability and wind shear and have improved representation of frontal waves and storms, and a more vigorous NIGW field would result. Komori et al. (2008) show NIGWs, with vertical velocity signatures twice as large as those shown here in NEMO, to occur in a coupled ocean–atmosphere model with a T239 resolution (50 km) atmosphere and a coupling period of 20 min. It is not known how the component ocean model itself would compare with NEMO given the same surface fluxes. As the ocean resolution is increased the Gulf Stream becomes more energetic and there is better extension around Cape Hatteras. Eddy activity is increased in this region, which coincides with the broken field of waves in Fig. 5a. Preliminary investigations with a 1/12th° version of the same model indicate that the waves propagating away from this region look very similar to those presented here. An important result is that the AMOC variability due to the NIGWs is found to exist in the residual term Wres when the model AMOC is decomposed into its component parts. This term is not measured by the RAPID observational array at 26.5°N, and therefore it would not be captured by the existing array design, even if higher temporal resolution were made available. However, the signature of the NIGWs will be captured by current meters, and possibly by T and S sensors. Below 1000 m temperature anomalies due to the NIGWs in NEMO are of the order 0.01 °C, and salinity anomalies are of the order 1  103 psu. Values higher than this, by about 1 order of magnitude can be found above 1000 m, where the gradients in T and S are stronger. The T and S anomalies are density compensating. Current meters are able to detect NIGWs (Alford, 2003), and data have also shown that NIGWs can have large spatial coherence (Thomson and Huggett, 1981). Satellite altimetry has been used to show increased NIGW activity in the western half of the North Atlantic (Glazman and Cheng, 1999) which is similar to what we find in our simulations (Fig. 5). However, despite such findings it is not clear whether the waves that exist in the real ocean are as spatially coherent as those seen in the model, whether they can travel over such long distances as suggested in our simulations, and indeed whether the waves in the real ocean exist so prevalently. Coupled ocean–atmosphere models have started using high (eddy-permitting) resolutions for the ocean (Komori et al., 2008). These ocean models are comparable to the NEMO ocean model discussed in the present study, which means that NIGWs are ubiquitous in high resolution coupled models used for weather forecast and climate predictions. This further highlights the need for us to understand the nature and impact of such waves both in the real world and in numerical models. The ubiquity of NIGWs in high resolution ocean models and their imprint on the AMOC highlights the possible importance NIGWs could have in the horizontal and vertical redistribution of wind energy throughout the ocean. The kinetic energy spectrum of the ocean is dominated by internal waves with near-inertial frequencies (Ferrari and Wunsch, 2009) and dissipation of this

A.T. Blaker et al. / Ocean Modelling 42 (2012) 50–56

energy throughout the global ocean results in mixing. Mixing induced by NIGWs could contribute to the distribution of tracers (e.g. temperature, salinity, carbon) in the ocean (Alford, 2003), which means that their imprint might not be confined to near inertial timescales but that they could also affect the mean state of the ocean. Acknowledgements This work was part of the NERC funded RAPID VALOR Project (NE/G007772/1) and Oceans2025. Data from the RAPID-WATCH MOC monitoring project are funded by the Natural Environment Research Council and are freely available from www.noc.soton. ac.uk/rapidmoc. We enjoyed discussions with Vladimir Ivchenko as well as helpful comments from Steven Griffies, and two anonymous reviewers. The authors wish to acknowledge use of the Ferret program for analysis and graphics in this paper (http://www. ferret.pmel.noaa.gov/Ferret/). Appendix A. Near inertial gravity (Poincaré) waves Near inertial gravity waves, sometimes also called Poincaré waves, are a family of gravity waves which are subject to the effects of Earth’s rotation. The dispersion relation for NIGWs can be obtained by starting from the shallow water equations derived using the hydrostatic approximation,

@u @g  f v ¼ g ; @t @x @v @g þ fu ¼ g ; @t @y

ðA:1Þ ðA:2Þ

where u and v are the zonal and meridional velocities, x and y are the zonal and meridional coordinates, g is the acceleration due to gravity and g is the deviation of the free surface. The continuity equation is

  @g @u @ v ¼ 0; þ þH @x @y @t

ðA:3Þ

where H is the ocean depth. The problem is made simpler by aligning the y-axis with the wave crests so that @/@y = 0 and assuming f and H to be constant. If we also ignore the effects of boundaries (A.2) becomes

@v ¼ fu @t

ðA:4Þ

and (A.3) becomes

@g @u ¼ H : @x @t

ðA:5Þ

Taking @/@t of (A.1) to get

@2u @v @ @g f ¼ g @x @t @t @t 2

ðA:6Þ

and substituting (A.4) and (A.5) gives

@2u @2u 2  f u ¼ gH : @x2 @t 2

ðA:7Þ

Now if we try a wave solution of the form u = uoexp[i(kx  xt)] we find the variation in x and t cancel and we get the dispersion relation,

x2 ¼ f 2 þ c2 k2 ;

ðA:8Þ

where x is frequency of the wave; f = 2Xsin/ is the latitude (/) 5 1 dependent Coriolis parameter, with being pffiffiffiffiffiffiX = 7.292  10 s the rotation rate of the Earth; c ¼ gH is the gravity wave speed;

55

and k represents the horizontal wave number. A similar solution can be found for internal (baroclinic) waves. The waves have a horizontal scale which is large compared with the depth. Group speed is fastest for short wavelengths, becoming increasingly slower for longer wavelengths, whilst phase speed becomes increasingly fast with longer wavelengths. At infinite wavelength inertial oscillations occur, and the waves do not propagate. The energy in NIGWs, as with most systems in the ocean, is concentrated in the lower frequencies; the lowest possible frequency for NIGWs being the inertial frequency. As noted by Anderson and Gill (1979), the energy associated with the waves, whilst near inertial at a given latitude, can have frequencies substantially higher than inertial by the time it has propagated to lower latitudes. For further reading, an excellent discussion on the properties of these waves can be found in Gill (1982).

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