Journal of Oceanography, Vol. 60, pp. 893 to 904, 2004

Application of a Two-Way Nested Model to the Seamount
Problem
H ISASHI HUKUDA1 and XINYU GUO 2,1*
1

Frontier Research System for Global Change,
Showamachi, Kanazawa-ku, Yokohama 236-0001, Japan
2
Center for Marine Environmental Studies, Ehime University,
Bunkyo-cho, Matsuyama 790-8577, Japan(Received 29 March 2002; in revised form 7 November 2003; accepted 8 December 2003)

A two-way nested model has been constructed and applied to the idealized ocean wherea uniform mean flow impinges on the isolated Gaussian-shaped seamount and produces two eddies (cold and warm) in the depths. The performance of the nested model
has been evaluated subjectively and objectively. Both subjective and objective analyses confirm the traditional view that the nested model can well capture the performance of isolated eddies. Objective analysis, however, reveals some quantitatively important features of a two-way nested model. One is penetration of improved features
into the coarse domain and another is the deterioration of mean flow field inside thenested area, neither of which is clear from subjective analysis. With successful application of two-way nested model to the seamount problem, we expect that such a nested
model will also be applicable to other oceanic phenomena, particularly to some coastal
problems whose time scale is short and where the topographic effects are dominant.

Keywords:⋅ Two-way nested
model,⋅ flux-conservation,⋅ seamount problem,⋅ eddy behind theseamount,⋅ objective analysis,⋅ sigma coordinate
model,⋅ message-passing
model.

to its meteorological applications, there are not many
examples of two-way nested ocean models. Spall andHolland (1991) developed a TNM and applied it to abarotropic modon and a baroclinic vortex in a flat ocean.
Fox and Maskell (1995) extended their methods to an
ocean with topography and attempted further refinementsof the grid in the vertical direction. The application of
their TNM to a front was successful (Fox and Maskell,
1996). It must be noted that all of these TNMs are basedon a z-coordinate model. Using on a sigma-coordinate
model (Blumberg and Mellor, 1987), Oey and Chen (1992)
developed a TNM to simulate meanders and eddies in the
Norwegian Coastal Current. Ginis et al. (1998) applied
the two-way nested scheme for hurricane prediction(Kurihara et al., 1979) to a layer ocean model and simulated the response of tropical Pacific Ocean to winds.
The use of TNM to study the coastal waters near orin Japan would be desirable. For example, there are many
narrow straits in the Seto Inland Sea that frequently induce topographic eddies with the passage of tidal current. The topographic eddies definitely affect the surrounding wide sea areas (Nada in Japanese) but the straits
are so narrow that it is unreasonable to completely resolve them in a model including the Nadas. The local topography is also important to the Kuroshio. For example, Endoh and Hibiya (2001) have shown that the interaction between the Kuroshio and Koshu Seamount, a

1. Introduction
The nested model is a useful means to resolve local
features that may not be resolved by a coarse grid model
alone. The merits of using such a model are clear. One
merit is that it gives a tool to resolve a local feature without using a fine grid everywhere, which involves a considerable sacrifice of computer time and memory. Anothermerit comes from a physical consideration; e.g., more heatis transported by a typhoon to the latitude at which it isfound by using a nested model rather than using a coarse
grid everywhere.
There are two kinds of nested models. One is a oneway nested model, in which the coarse grid model provides boundary conditions to the embedded fine gridmodel. The other is a two-way nested model (TNM), in
which the fine grid model not only obtains boundary conditions from the coarse grid model, but also feeds its results back to the coarse grid model. Thus, a TNM may be
regarded as an attempt to assimilate fine grids into coarse
grids with some analogy to a data assimilation model inwhich data plays the role of the fine grid solution.
In principle, TNM is more reasonable than a oneway nested model. However, in practice, TNM is moredifficult to use than a one-way nested model. Compared* Corresponding author. E-mail: guoxinyu@dpc.ehime-u.ac.jp
Copyright © The Oceanographic Society of Japan.

893

seamount south of Japan, plays a crucial role in deciding
the path of Kuroshio.
The above TNMs, except for that of Ginis et al.
(1998), do not conserve the flux strictly across the nested
boundary. Usually a conservative scheme is more complex than a nonconservative one. Furthermore, it is notclear that a conservative scheme can give a better result
than a nonconservative one (Spall and Holland, 1991)
because in a nested model the coarse grid model cannot
resolve the solution of the fine grid model and a conservative scheme enforces the coarse grid fluxes on the finegrid solution and may propagate the erroneous coarse gridfield more quickly into the fine grid region (Spall andHolland, 1991).
For
relatively
short-term
integrations,nonconservative boundary conditions are acceptable(Spall and Holland, 1991; Fox and Maskell, 1995). But,
as noted by Spall and Holland, this conclusion depends
on the model application. The topographic eddies in the
Seto Inland Sea are produced by the ambient tidal current. The trapped abyssal anticyclone above Koshu
Seamount results from the ambient Kuroshio (Endoh andHibiya, 2001). These show that the interaction of ambient current with topography is important. Although theeddies reported by Spall and Holland (1991) or the fronts
described by Fox and Maskell (1995) could induce a background current, the ambient current itself is not imposedin their models. Therefore, the following question remains
to be answered: what is the performance of a TNM with
nonconservative boundary conditions in the presence ofan ambient current?
In this paper we address such a problem using theTNM that has been constructed based on POM (Mellor,1998) and PVM (Geist et al., 1994). The seamount problem described in Chapman and Haidvogel (1992) and
Mellor et al. (1998) gives an ideal situation for such a
study because it involves both mean flow and isolated
eddies for which the validity of the TNM without flux
conservation may be examined. Another difference of thisstudy from those of Spall and Holland (1991) and Fox
and Maskell (1995) is that we use a sigma coordinate
ocean model.
The present paper describes the TNM in Section 2.
In Section 3 we revisit the seamount problem and apply
our nested model to examine its performance subjectively.
In Section 4 we give an objective method to evaluate theerror of a nested model. A conclusion is given in Section
5.
2. Description of Model
2.1 Basic ocean model
The basic ocean model we use is a primitive equation model known as POM (Mellor, 1998). This model is894

H. Hukuda and X. Guo

Fig. 1. Nested grid structure: interpolation is done along the
dynamic interface. Feedback is made on all grid points
within the feedback interface.

briefly outlined as follows. It adopts Arakawa C grids(Arakawa and Lamb, 1977), uses the sigma coordinate as
the vertical coordinate (Phillips, 1957) and utilizes themode split method in which external and internal modes
are solved with different time steps (Simons, 1980). It
also contains the Mellor-Yamada turbulent closure modeland Smagorinsky nonlinear viscosity. Although it is applicable to the orthogonal curvilinear spherical coordinate system, we are only concerned with the rectilinear fplane system in this study.
2.2 Nesting procedure
In the present nested model, a coarse grid model(CGM) and a nested fine grid model (FGM) are both
treated as independent models, except for the communication parts in which two models exchange each variables
for interpolation and feedback. Therefore, we actually run
two models in parallel. The PVM message-passing modelis used for communication. In this model the daemon process called PVMD plays the role of a message-passer. We
have observed that latency (i.e. cpu time required for themessage-passing process) was quite small (a few percentof the total cpu time) for our Unix workstation. The
present model, however, is coded so as to use a minimum
CFL time step required by the finest grid model and thisis necessary in order to simplify the nesting code. The
nested grid structure is depicted in Fig. 1 for a grid ratioof 1:3.
a. Interpolation
The interpolation method used here is based on theidea that the error caused by interpolation may be reduced

φi = εΦ I + (1 − ε )φi −1(1)

where φi and φ i–1 denote an FGM variable at the boundary grid and at the inward grid, respectively, and ΦI is thevalue at the auxiliary point that is an intermediate pointof the CGM grids outward from the boundary, as interpolated from the adjacent CGM variables along the dynamic
interface. Let δ x and δX be the grid-width of the fine andcoarse mesh, respectively. Referring to Fig. 1, we can set
ε = 2δ x/(δx + δX) for the temperature and tangential velocity points, and ε = 1 for the normal velocity points. It
is important to note that as POM uses Smagorinsky
nonlinear viscosity (AM), we need to use (1) to determine
the value of AM at the boundary grids of FGM.
b. Feedback
The feedback is incorporated into the model by applying Spall and Holland’s (1991) method. In the case of
a 1:3 grid ratio (see Fig. 1), this is equivalent to using
9

AI Φ I = ∑ aiφi .
i =1

(2 )

In (2) one coarse grid of area AI is composed of nine finegrids of area ai with the constraint that AI = ∑ i9=1 ai (Iand i being the CGM and FGM grid point index, respectively). Equation (2) is applied to all CGM prognostic
variables within the feedback interface at every time stepof external and internal modes.
c. Topography
The topography is handled as follows. First the water depths of FGM and CGM are smoothed using the volume-conserving filter described in Mellor (1998). However, the smoothing is overridden for the CGM depth inside the feedback interface where both the FGM and CGM
domain overlap, for which (2) is used to determine thedepth. The depths on the FGM boundary grids are interpolated from CGM depths using (1). We should note that
this prescription of depth is valid only if the nested boundary is located on a relatively flat depth so that the crosssectional areas from FGM and CGM are of equal magnitude. For the seamount problem discussed below, we

m

Depth

by using the values on nearest grids. Thus, the values at
the FGM points along the dynamic interface are determined by simple linear interpolation. This is done firstalong the tangential direction using two adjacent CGM
variables to decide the value at the outward grid nearest
the dynamic interface (auxiliary point) and then along thenormal direction using one CGM variable at the auxiliary point and one FGM variable located inward from theinterface by one grid. For the normal velocity, the interpolation in the normal direction is redundant. This operation is written asm

X

Fig. 2. Seamount model: Solid (dashed) line shows the depthof FGM (CGM). The initial temperature profile of the modelis also indicated with a dashed line.

chose the nested boundary to enclose the seamount andthe flux error caused by the depth is almost negligible.
Thus, since no temporal interpolation error occurs in our
TNM which uses the same time step for CGM and FGM,
the sole flux error comes from the spatial interpolation
error of the model field along the dynamic interface.
3. Experiments with the Seamount Model
3.1 A seamount model
The seamount model is taken from the default setup
in POM. A brief description of the model is given below.
The forcing is given by a barotropic mean flow flowing
eastward. The ocean, with a size of 500 km × 400 km, isa channel enclosed by rigid walls at the northern andsouthern boundaries and has an inflow and outflow atwestern and eastern open boundaries, respectively. We
specify a steep seamount with a Gaussian profile{h( x, y) = h∞ 1 − δ exp − ( x − xc ) + ( y − yc )


2

2

} r 
2a(3)

where h∞ = 4500 m, r a = 25 km and (x c, yc) denotes the
position of the seamount center. The parameter δ measures the seamount height relative to the oceanic depth
and is fixed at δ = 0.9 in this study.
The ocean has an initial stratification in which only
temperature varies with the depth z as
T = Ta + Tb exp[γz ]( 4)where T a = 5°C, T b = 15°C and γ = 10–3 m–1 (see Fig. 2).
The temperatures at the two open boundaries are determined by upstream differences. Salinity is simply fixed
at a constant value (35 psu). Moreover, heat and momentum fluxes at the sea surface are assumed to be zero. Thus,

Application of a Two-Way Nested Model to the Seamount Problem

895Table 1. List of experiments: the figures in parentheses refer to CGM to FGM values in order and ( δTe, δ Ti ) refer to the external
and internal mode time step.
No.

Grid type1
2
3

SGM low
SGM high
Doubly nested

Grid spacing (km)45Triply nested
Movable nest

8.1
2.7
(8.1, 2.7)
(8.1, 0.9)
(8.1, 2.7, 0.9)
(8.1, 2.7, 2.7)
(8.1, 2.7, 2.7)
(8.1, 2.7, 2.7)

CFL time (δTe, δTi sec)

Grid number (I × J)
65 × 49
195 × 147
(65 × 49, 50 × 38)(65 × 49, 146 × 110)
(65 × 49, 50 × 38, 38 × 38)(65 × 49, 38 × 38, 38 × 38)(65 × 49, 50 × 50, 50 × 50)(65 × 49, 62 × 62, 62 × 62)(9, 270)(3, 90)(3, 90)(1, 30)(1, 30)(3, 90)(3, 90)(3, 90)

5.28

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SGM-h
A 32
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A 33
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Fig. 3. Temperature contours at 4000 m of Case SGM-l (left), Case A32 (middle), and Case SGM-h (right). Contours are drawn at
every 0.024°C.

the only forcing is the uniform eastward flow that impinges on the seamount. The boundary conditions at eastern and western ends are taken from the default setup in
the POM’s subroutine BCOND. Both CGM and FGM startfrom the same initial conditions with zero elevation, uniform eastward flow of 0.2 ms–1, and the temperature specified in (4).
Huppert and Bryan (1976) and Chapman and
Haidvogel (1992) have described the solution of the above
seamount problem in detail. Basically, we obtain a similar solution. As soon as the model starts, two eddies with
opposite signs of vorticity are formed at the seamount.
896

H. Hukuda and X. Guo

The upwelling on the upstream side of the seamount compresses water columns and induces an anti-cyclonic (cold)
eddy. The downwelling on the downstream side stretches
water columns and induces a cyclonic (warm) eddy. These
two eddies co-rotate clockwise around the seamount.
While radiating inertia gravity waves, two eddies interact and reach geostrophic balance after a couple of days.
After Day 2~3 the advective process becomes dominant.
The warm cyclonic eddy is detached from the southeastern foot of the seamount and the mean flow advects it to
the east. Meanwhile, the cold anti-cyclonic eddy remains
trapped around the seamount.

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Fig. 4. As Fig. 3, but for temperature at 1000 m. Contours are drawn at every 0.05°C.

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Fig. 5. As Fig. 3, but for elevation. Contours are drawn at every 0.08 m. Seamount topography is also shown.

3.2 Experiments with the nested model
Table 1 summarizes the experiments conducted inthis study. Two control experiments (Exp. 1, 2) with two
single grid models (SGM), each having a low resolution(8.1 km) or a high resolution (2.7 km) everywhere, wererun prior to the nested model experiments. All models
use the same eleven σ levels given by σ = 0, –.018,–.036, –.071, –.143, –.286, –.429, –.571, –.714, –.857,–1 and the same horizontal momentum and heat diffusion parameters, HORCON = 0.2 and TPRNI = 1 (seeApplication of a Two-Way Nested Model to the Seamount Problem

897

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Fig. 6. Temperature contours at 50 m of Case A32 (left), Case A33 (middle), and Case A92 (right). Contours are drawn at every
0.1°C

Mellor (1998) for definition). The nested model was run
stably for a month. However, because the flow reaches anearly steady state after 15 days, the period for which thewarm eddy generated at the seamount flows out of theeastern end, only results until Day 15 are presented below.
In the remainder of this section we describe two kindsof experiments using the TNM; the first one only uses anested area fixed over the seamount (Case A) while thesecond covers each of cold and warm eddies by using
multiple nests (Case AI). We compare the performanceof our nested model with that of two SGM results.
a. Case A
The doubly nested model where the nested boundary is located on a flat bottom surrounding the seamountis standard and will be referred to as Case A32 (hereafter,a subscript refers to the grid ratio and a superscript refers
to the degree of nest). Since the two eddies have a strong
vertical structure with bottom intensification (Chapman
and Haidvogel, 1992), we first show the evolution of temperature contours at 4000 m depth (Fig. 3), where theTNM result (labeled nest) is compared with two SGM
results (labeled coarse and fine). We can see how the horizontal grid size affects the eddy evolution. The cold eddy
surrounding the seamount is more diffusive on the coarse
grid, but less diffusive on the fine one. The nested model
can resolve this feature well. On the other hand, the warm
898

H. Hukuda and X. Guoeddy flowing away from the seamount is much weakeron the coarse grid, but is enhanced on the fine grid. The
nested model clearly fails to reproduce this feature simply because the resolution is not enough there. The horizontal resolution also affects the evolution of the warm
eddy in such a way that the eddy structure is much more
self-organized in the high-resolution model result.
Fairly speaking, Case A32 seems to show a reasonable result and confirms our previous knowledge of TNM:
a mesoscale feature inside the nest can be resolved reasonably well, but other features outside the nest are only
poorly resolved; thus, as expected, it yields a hybrid result of CGM and FGM characteristics. Case A32, however, clearly fails in reproducing the warm eddy structure as calculated by the high-resolution model. This failure prompted us to use multiple nests so as to cover eachof the eddies generated around the seamount. Before doing so, however, we want to look at the model performance at shallower depths.
How well does the same nested model perform at
the shallow level? Figure 4 shows the evolution of temperature contours at 1000 m depth on the coarse, nested
and fine grids. The nested model result is taken from thesame outputs as in Case A32. We see that little noise appears across the nested boundary and its performancecompares well with that at the deep level.

Y

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Fig. 7. Temperature contours in Case AI. Contours are drawn at every 0.025°C. Nest size increases in order from left (Case AIs),
middle (Case AIm) to right (Case AIl).

The final comparison in Case A32 is made for theelevation and is shown in Fig. 5. The northward declineof sea level implies the presence of an eastward flow,which is disturbed by the seamount topography. The sea
level of a nested model (middle) compares well with thatof a high-resolution model (right), although the difference among three (coarse, nest and fine) models seems tobe small. Thus, subjective judgment of the nested model
performance for the mean flow field is less critical thanthat for the eddy field.
A triply nested model, which is a simple extensionof the doubly nested model, may be used to resolve a finestructure near the surface. This case amounts to setting
one more nest in Case A32 and will be referred to as Case
A33. For this high-resolution grid, the control experimentthat uses a SGM with a grid spacing of 0.9 km is not feasible. Instead, we decided to compare this result with thatof the doubly nested model with a high grid ratio of 1:9(Case A92). This is permissible because, with a high gridratio, the doubly nested model is less subject to the influence from the nested boundary, so it will be regarded as
an approximate control experiment for the innermost areaof the triply nested model.
Figure 6 compares temperature contours at 50 mdepth between the triply nested model (middle) and two
doubly nested models, one with a high grid ratio of 1:9(right) and other with a low grid ratio of 1:3 (left). We
see that the eddy at the seamount summit becomes finer
as resolution increases from 2.7 km (left) to 0.9 km (mid-dle and right). We also see that both high-resolution models reproduce a similar pattern, but the triply nested model(middle) is more susceptible to the boundary effect compared to the doubly nested model (right), as mentioned
above.
The dominant stationary eddies appear at the
seamount summit and may be recognized as the Taylor
caps described by Chapman and Haidvogel (1992). According to their experiments, the Taylor caps remaintrapped around the seamount only at low Rossby numbers. With a speed of u = 0.2 ms–1, the present case gives
Ro = 0.08 and satisfies Chapman and Haidvogel’s (1992)
criterion for Taylor caps to be trapped near the seamountsummit. Thus, Fig. 6 seems to give further credit to the
TNM performance.
b. Case AI
Since the warm eddy moves downstream as time
passes, it is necessary to introduce a movable fine meshto follow and resolve the warm eddy. The movable mesh
was incorporated in the present model by following themethod of Kurihara et al. (1979). We refer the reader tothat paper for details of the movable mesh technique. The
only thing to mention here is to replace (1) by ∂Φ  ∂Φ 
φi = Φ 0 +  0  ( xi − X0 ) +  0  ( yi − Y0 )
 ∂Y 
 ∂X 

(5)where (xi, yi) and (X0, Y0) are the coordinates of the fine

Application of a Two-Way Nested Model to the Seamount Problem

899

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Fig. 8. As Fig. 7, but for elevation. Contours are drawn at every 0.08 m. Seamount topography is also shown.

and coarse grids in the area that was newly created whenthe mesh moved. The apparent characteristic of (5) is that
adding (5) for φi over one coarse grid leads to the value
at the CGM grid (Φ0) so that a conservation law holds
during the mesh movement.
Initially two nested areas are placed to cover eacheddy and the nest covering the warm eddy is allowed tomove with the eddy. Figure 7 shows three cases simulated by the multiply nested model for which the nest size
was increased slightly from Case AIs to Case AIl. Generally speaking, Case AIs seems to give enough size to reproduce two eddies well. The similarity of Case AIs tothe high-resolution SGM in Fig. 3 is remarkable: the detached warm eddy structure is well reproduced in thismodel. It also appears that as the nest size increases, theresolved eddy field becomes closer to the high-resolution result.
Figure 8 compares the elevation field in Case AI.
Subjectively, we can see little noise in the sea level that
could be caused by the presence of two nested fine meshareas. There remains, however, a question of how the flux
error affected the performance of our TNM when thenested area is placed in the mean flow field. We evaluate
these errors as well as the errors in Case A in the next
section. Because the experimental results have so far beeninterpreted only according to a subjective judgment of
the simulation results, it is not certain that the judgment
is free from personal bias. The evaluation of the model
performance by objective analysis makes this point clear.
900

H. Hukuda and X. Guo

4. Objective Analyses
4.1 Subjective versus objective
In the previous section we judged the nested model
performance based on a visual comparison of the resolved
features with those of the SGM. Such a subjective judgment certainly gives us a base to verify whether the nested
model performed well. It is, however, more desirable toseek another way in which the model performance can beobjectively evaluated. This may be more convincing thanasking the reader to compare wiggles in figures.
We consider two methods to evaluate the model performance quantitatively. These are the conventional methods used to see the accuracy of TNM: one in which thegeometric quantities of eddies are estimated and the otherin which the root mean square (RMS) errors of the field
variables are estimated.
4.2 Estimate of eddy properties
Figure 9 compares the area occupied by the cold eddy
(T < 5.28°C) among the coarse, fine and nested model asin Case A 32 at 4000 m. We can clearly see the similarityof the curves between the fine and nested model and thedissimilarity of these to the curve of the coarse model.
Furthermore, we see that the nested curve lies between
the coarse and fine model curves, fitting between them.
Figure 9 thus quantitatively reconfirms our subjective
remarks on the nested model performance.
Figure 10 compares the maximum temperature of the

Area

y

m

m

x
Time

Fig. 11. Trajectories of the warm eddy center in Case SGM-l(solid circles), Case AIs (open squares), and Case SGM-h(open circles).

(Day)

Tmax

Fig. 9. Time evolution of the cold eddy area with T < 5.28°C at
4000 m in Case SGM-l (solid circles), Case A 32 (opensquares), and Case SGM-h (open circles). Area is made nondimensional by dividing by the seamount area at 4000 m.

Time

(Day)

Fig. 10. Time evolution of the warm eddy’s maximum temperature in Case SGM-l (solid circles), Case AIs (opensquares), and Case SGM-h (open circles).

warm eddy among the coarse, fine and the movable nested
model as in Case AIs. The curves of the fine and nested
model are very close while the coarse model gives muchsmaller maximum values. Again, the nested curve fits
between the coarse and fine model curves and reconfirms
our subjective remarks.
These objective analyses clearly show that the eddy
properties associated with the temperature field are well
predicted by the TNM. This success is likely due to theisolated feature seen in the temperature contours at 4000m-depth as TNM has a good capacity to resolve such anisolated phenomenon.
However, Fig. 11 shows some counterproductive
performance of the TNM where the predicted track of thewarm eddy center is less satisfactory than that of thecoarse model. In their study of typhoon simulation, Ley
and Elsberry (1976) showed that the track of Typhoon

Irma is closer to the coarse model result than to the nested
model one. It is interesting to see the similarity between
their study and the present one, although the two studies
treat very different subjects. The reason for this deterioration is likely due to the change in a mean flow that affects the movement of an isolated eddy. We examine thisconjecture in the next subsection.
4.3 Estimate of root mean square errors
The reason why the mean flow is changed by the useof TNM is interesting. To estimate the change quantitatively, we must devise a measure of error in the mean
flow and the eddy field. For this purpose we introduce
the root mean square (RMS) error of the nested model
variable relative to the true solution that is given approximately by the fine model result. To see the effect of feedback from FGM to CGM, we calculate RMS errors separately for the inside and outside region of a nested area.
Let us denote these errors as εi and εo, respectively. Thus,
we define

εi =

εo =

∑ (Φ n − Φ f )2Ni∑ (Φ n − Φ f )No∑ (Φ c − Φ f )

2

(6 )

Ni
2

∑ (Φ c − Φ f )

2

No

(7)

where Φn, Φf, Φc are the quantities of the nested, fine andcoarse grid model, respectively; Ni and No are the numbers of the grid points on the inside and outside of thenest, respectively; the sum in (6) and (7) is taken over the
CGM grid points lying inside and outside the nest, respectively. As the value of the FGM variable, we used anine-point averaged value of Φf defined on the CGM grid.
Note that the RMS error of the nested model is nor-

Application of a Two-Way Nested Model to the Seamount Problem

901

m

malized with that of the coarse model. Thus if εi, εo < 1,
the nested model performance improves on the coarse
model, while if εi, εo > 1, the nested model performanceis inferior to the coarse model. The RMS errors were calculated for elevation ( η), temperature (T), and zonal andmeridional velocity components (U, V) using (6) and (7).
a. Case A
We first look at the RMS errors in Case A32. Figure
12 shows these errors at 4000 m plotted against time at
the interval of 2.5 days. We see that all errors inside thenest (marked by open circles) are less than unity for thisshort-time evolution, implying that the TNM improvedon the coarse model. An interesting feature in Fig. 12 isthat the temperature error outside the nest also decreases
from unity for periods between Day 2.5 and Day 15, whichmeans that the improvement of temperature inside the nest
affected the temperature outside the nest and reduced theerror there. This is a merit of having a TNM where theoutside CGM solution is influenced by the nested FGM
solution. Another feature to be noticed in Fig. 12 is thatthe sea level error inside the nest (<0.9) is less improved
compared with the temperature error in the same region(<0.3). This is caused by the error associated with interpolation of the elevation, which increases due to the sea
level gradient along the nested boundary associated witha mean flow field.
To make the above interpretation more concrete, we
calculated the RMS errors that show vertical profiles averaged over whole time periods (Fig. 13). We see that a902

H. Hukuda and X. Guomm

Fig. 12. Root mean square errors at 4000 m for Case A32 plotted against time for regions of inside (open circles) andoutside (solid circles) of the nested area.

Fig. 13. As Fig. 12, but plotted against depth except for theelevation.

greater improvement is attained for temperature (<0.2)
than elevation (<0.9). The velocity errors are of intermediate values (<0.8) between them, but closer to the elevation error. This is because the mean flow is barotropic
and the sea level gradient is a main component in determining the flow.
b. Case AI
It is of interest to extend the above analysis to the
movable nest case. Figure 14 shows the RMS errors forCase AIs. Because this case uses two nests, one fixed andthe other movable, two kinds of RMS errors are introduced for each eddy. In this case the elevation error increased from unity, implying that the TNM performance
deteriorates below the coarse model. However, the temperature error still remains smaller than unity and improves on the coarse model. Generally, the temperatureerror inside the movable nest is larger than that inside thefixed nest.
In Fig. 15 the RMS errors averaged over the whole
periods are plotted against the depth. We see that nearsurface zonal velocity (U) errors for the warm eddy become large, but decrease as depth increases. This error
may be explained by the pressure gradient error associated with the large elevation error and by the fact that theeddy traced by the movable mesh is at 4000 m and notnear the surface. The three-dimensional eddy structure
generated behind the seamount is thus quite difficult tofollow by a simple two-dimensional nesting technique.
The reason for the deterioration of sea level in thiscase is attributable to two causes. One is the increase offlux error in Case AIs as the nested area doubled from

Table 2. RMS errors for the warm eddy averaged for time.

η
T at 4000 m

Case AIs

Case AIm

Case AIl

1.011
0.574

1.047
0.373

1.289
0.294

Table 3. RMS errors for the cold eddy averaged for time.

η
T at 4000 m

Case AIm

Case AIl

1.27
0.246

1.217
0.197

1.292
0.232

change in a mean flow, as shown in Fig. 11, leads to thelarger RMS error than the coarse model by changing thelocation of the warm eddy. It is not likely that the meshmovement over the seamount affects the flux error because the seamount topography is perfectly covered with
two fine nests. Thus, we can say that the presence of amean flow field, by geostrophy, generates the large gradient of sea level along the nested boundary and favors
large interpolation errors.
Further calculations of RMS errors in Case AIs to
Case AIl show that as the size of the nested area increases,
the temperature error decreases whereas the elevation
error increases for the warm eddy (see Table 2). This isbecause the warm eddy structure is well isolated in thecurrent problem and the TNM usually improves such a
local feature with increasing nest size. The sea level contours, on the other hand, are not localized in the present
case and have a wide coverage, straddling the whole nest,
so the increase of the nest size will increase the error.
A similar trend of RMS errors is seen when we compare Case AIs to Case AIl for the cold eddy (Table 3), but
the values this time show a less definite trend than that of
the warm eddy.

m

m

m

Fig. 14. Root mean square errors at 4000 m for Case AIs plotted against time for regions of nested cold eddy area (opencircles), nested warm eddy area (open squares), and outside of the nested areas (solid circles).

Case AIs

Fig. 15. As Fig. 14, but plotted against depth except for theelevation.

Case A. Because the flux error is proportional to the sizeof a nested area for the TNM without flux conservation,
we expect larger flux errors in Case AIs than Case A, as
well as in Case AIl rather than Case AIs. The resulting

5. Conclusion
By constructing a TNM that does not have the fluxconserving property and applying it to the idealized
seamount problem, we have studies the applicability ofsuch a nested model to the case when an ambient current
produces isolated eddies. The nested model was run for ashort time of 15 days and its performance has been evaluated using both subjective and objective analyses. Subjective analysis is not new as it reviews our knowledgeof the TNM that is well applicable to the mesoscale, isolated patterns. Objective analysis using the root meansquare error was then introduced to estimate the error ofa nested model both inside and outside of the nest. This
analysis gives us two new features. One is a penetration

Application of a Two-Way Nested Model to the Seamount Problem903

of the improved features from fine to coarse grid areas.
Another is deterioration of the sea level error associatedwith the mean flow field relative to the temperature errorof the eddy field, which is not clear at all from the subjective analysis only. Therefore, with the objective analysis method proposed in this study, one may attain deeper
insight into the problems related to the nested model.
The present study apparently shows both the meritand demerit of the two-way nested model without flux
conservation. The model has proved to be useful for simulating the isolated feature of an eddy field observed in
the seamount problem. However, its performance is notvery much improved for simulation of the mean flow field
because our model does not conserve the fluxes acrossthe nested boundary, in common with most past models.
The presence of mean flow produces, by geostrophy, thelarge gradient of sea level along the nested boundary andthis leads to the interpolation error of flux. This error inturn changes the mean flow field and thus affects the eddy
position advected by that mean flow.
This kind of error might be more serious when wetreat a more general situation characterized by a steep
continental slope and baroclinic western boundary currents like the Kuroshio. Then we must choose the nested
area carefully in order to circumvent the large gradientof the scalar variables such as depth, temperature and sea
level, which may increase the interpolation error. Accordingly, the size and position of the nested area may not bechosen from economic point of view only, so as to eliminate the flux error as completely as possible. However,
the TNM must be a useful tool to study the tide-induced
topographic eddies in the Seto Inland Sea because mostof the straits there are surrounded by relatively flat Nadas
and the eddies have a short lifespan.
Acknowledgements
This work was done as part of the Japan Coastal
Ocean Predictability Experiment (JCOPE) project, supported by Japan Marine Science and Technology Center.
We thank Dr. Takashi Kagimoto and Dr. Jim Kohl foradvises and supports in computational aspects, and Ms.
Tomoe Mikami for drawing figures. Comments of tworeviewers were helpful in improving the manuscript.
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