Properties and evolution of a submesoscale cyclonic spiral

Abstract. The evolution of a submesoscale cyclonic spiral of 1 km in diameter is simulated with ROMS (Regional Ocean Modeling System) using 33.3 m horizontal resolution in a triple-nested configuration. The generation of the spiral starts from a dense filament that is rolled into a vortex and detaches from the filament. During spin-up, extreme values are attained by various quantities, that are organized in single-arm and multi-arm spirals. The spin-down starts when the cyclone separates from the filament. At the same time, the horizontal speed develops a dipole-like pattern and isotachs form closed contours around the vortex center. The amplitudes of most quantities decrease significantly, but the instantaneous vertical velocity w exhibits high-frequency oscillations and more pronounced extremes than during spin-up. The oscillations are due to vortex Rossby waves (VRWs), that circle the eddy counterclockwise and generate multi-arm spirals with alternating signs by means of azimuthal vorticity advection. Experiments with virtual surface drifters and isopycnal floats indicate downwelling everywhere near the surface. The downwelling is most intense in the center of the spiral at all depth levels, leading to a radial outflow in the thermocline and weak upwelling at the periphery. This overturning circulation is driven by convergent near-surface flow and associated subduction of isopycnals. While the downwelling in the center may support the export of particulate organic carbon from the mixed layer into the main thermocline, the upwelling at the periphery effectuates an upward isopycnal transport of nutrients, enhancing the growth of phytoplankton in the euphotic zone.


Due to the small spatiotemporal scales, it is rather difficult to observe submesoscale spirals in situ. Therefore, the actual knowledge of the physical properties of the spirals is largely shaped by their signature at the sea surface detected by remote sensing techniques. The corresponding observations are mostly limited to images of proxy parameters such as sun-glint (Scully-Power, 1986), surface roughness (Karimova and Gade, 2016), phytoplankton (Fig. 1), or surfactants like oil slicks (D'Asaro et al., 2018). Namely, such snapshots allow the identification of sharp convergence lines ( Fig. 1) suggesting downwelling in 5 submesoscale fronts and spirals according to theoretical studies, but they cannot describe the formation and the decay of the spirals and the associated three-dimensional circulation patterns. Some more insight is provided by in-situ observations of D' Asaro et al. (2018) of a cyclonic vortex about 10 km in diameter, concerning the internal mass field, kinematic properties and the clustering and dispersion of surface drifters. The SubEx campaign of Marmorino et al. (2018) focused on a much smaller cyclonic eddy, the diameter of which was around 1 km. From airborne infrared measurements and nearly simultaneous 10 observations with ADCP (Acoustic Doppler Current Profiler) and a Towed Instrument Array, kinematic quantities such as horizontal velocity, vorticity, horizontal strain, and divergence were estimated. In the framework of the same experiment, Ohlmann et al. (2017) computed divergence and vorticity distributions from surface drifters. To our knowledge, only the latter 3 citations provide concrete findings about the properties of submesoscale cyclonic spirals, but the perceptions are still rather fragmentary. Before describing the evolution of a submesoscale spiral, we will therefore compile in the following some 15 information concerning the corresponding patterns in mesoscale, primarily cyclonic eddies. Namely, the spatiotemporal scales of such eddies are several magnitudes larger, but sometimes they exhibit striking similarities to submesoscale spirals.
A comprehensive observational study during the formation of a Southern Ocean cyclonic mesoscale eddy was conducted by Adams et al. (2017), using drifters, Seasoar (a towed, undulating underwater vehicle), and ADCP. While the drifters orbited the cyclone along its main front, cross-sections perpendicular to the drifters' paths were carried out simultaneously with the 20 other instruments. The collocated Lagrangian drifter velocities and the Eulerian ADCP measurements showed an asymmetric ringlike circulation with high horizontal velocities in the north and weaker velocities in the south. Furthermore, regions of diffluent and confluent cross-frontal velocities were identified. Upwelling was associated with diffluent and downwelling with confluent flow patterns. In another study, Buongiorno Nardelli (2013) described the evolution of a cyclonic mesoscale eddy in the Agulhas Return Current. The vertical circulation pattern was dominated by azimuthal oscillations, known as vortex Rossby 25 waves (McWilliams et al., 2003). In a purely numerical study, Gula et al. (2016) described in detail the structure of various quantities inside a cyclonic Gulf Stream eddy. Further information may be gained from investigations of anticyclonic mesoscale eddies. For instance, dipolar and multipolar appearences of the vertical velocity were observed by Barceló-Llull et al. (2017), and are described in the numerical study of Estrada-Allis et al. (2019). The complex structure of the vertical velocity field was subject of the investigations of Koszalka et al. (2009), Brannigan (2016 and Brannigan et al. (2017). Amongst others, they 30 revealed spiraliform patterns and closely spaced cells of either sign at the periphery of an anticyclone. Similar to Gula et al. (2016) in the case of a cyclonic eddy (see above), Zhong et al. (2017) investigated various quantities in an anticyclonic eddy, with special emphasis of the vertical circulation pattern.
In order to deepen our knowledge of the evolution and physical properties of submesoscale vortices, the Regional Ocean Modeling System (ROMS) is applied in a triple offline-nested configuration for a subregion of the Baltic Sea, located to the in the top layer on 24-28 June for R100 and R33. On 24 June (panels a, f), the fields of both nests look identical, hence the downscaling appears to be perfect. First noticeable differences between the nests appear on 25 June in the northwest sector.
The disagreements increase on 26 and 27 June, and additional disparities to the south of the spiral appear, but the spiral is not yet affected. Finally on 28 June, the R33 patterns deviate significantly from R100, and also the spiral exhibits differences; 25 in R100, its shape is more elliptical and the high-density core is better preserved in R33. Overall, the nesting procedure does apparently a good job.

Fronts, frontogenesis and frontolysis
In the framework of this article, the term "front" denotes density fronts, i.e. regions with enhanced gradients of the horizontal density ρ. A measure of the strength of a front is |∇ρ|. The frontal tendency F = d|∇ρ|/dt indicates whether a front intensifies (frontogenesis, F > 0) or weakens (frontolysis, F < 0). The enlarged view of the evolution of the surface density in Fig. 4a e reveals details that are not visible in Fig. 3f -j, e.g. the asymmetry of the filament (panel a) and the egg-shaped high-density 5 core of the spiral (panels b -e). The development of |∇ρ| (panels f -j) indicates that on 24 June, the strongest gradients of about 6 × 10 −4 kg m −4 are located at the western flank and close to the "head" of the filament in the north, while the gradients on the eastern flank are weaker. One day later, the maximum gradient has almost doubled and attains an all-time high of more than 11 × 10 −4 kg m −4 in the hook-shaped structure evolving later on into a spiraliform pattern. The strong enhancement of the density gradients during the first two days of the integration and the subsequent slackening is also reflected by the frontal 10 tendency (panels k -o) exhibiting intense frontogenesis of 59 × 10 −12 kg −2 m −8 s −1 and 64 × 10 −12 kg −2 m −8 s −1 in the red patches of panels l and m, respectively. However, at the same time, extreme weakening of up to ≈ −48 × 10 −12 kg −2 m −8 s −1 of the gradients takes place in the blue patches. Further analyses of the components of F (see Hoskins (1982); Capet et al. (2008); Onken et al. (2020a)) have shown that at this stage the major contribution to the frontogenetic tendency comes from Q h , the straining deformation of the horizontal velocity, while the vertical straining is the main driver of frontolytic processes. 15 The Q h pattern of 25 June (panel q) resembles closely that of Gula (2016, their Fig. 10f); this is the strong frontogenetic contribution on the upstream face of the spiral and the frontolytic sector at the tip of the hook.

Horizontal velocity
The total horizontal velocity V is the sum of the geostrophic and the ageostrophic velocity, (1)

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On 24 June (Fig. 5a), the northward progression of the filament is perceptible by |V | > 5 cm s −1 , where the maximum speed is close to 13 cm s −1 at the eastern flank of the high-density core (cf. Fig. 4a are found on 24 June at the head of the filament, and on 25 June inside the evolving spiral. Afterwards, the maximum values weaken to less than 2 cm s −1 and the ageostrophic speed reflects a multi-arm spiraliform vortex shape on 26 June. Moreover,   -(t) the ratio |V ageo|/|V geo| on 24 -28 June in R33. The color axis in (p) -(t) was limited by 100% because inifinitely high ratios occur for |V geo| = 0.
Vectors are drawn at 300-m resolution. All subplots are zooms and and approximately centered at the local density maximum. The dashed ruler in the northwest corner of each subplot represents a total horizontal distance of 1 km. the ageostrophic flow exhibits qualitative similarities with Fig. 10h in Gula et al. (2016), particularly the divergence at the tip of the hook. The horizontal distribution of the ratio between the ageostrophic and the geostrophic speed (panels p -t) reveals that the ageostrophic portion is large in regions where the geostrophic flow is extremely weak. For instance on 24 June, |V geo | is close to 0 cm s −1 in the northwest quadrant while |V ageo | is between about 1 cm s −1 and the extreme value of 2.7 cm s −1 (see above). The same is true for the spots marked in red on the following days, especially in the center of the spiral and at 5 the eastern boundary of the image in panel q. However, ratios between 40% and 60% are found in the filament on 24 June and in the hook-shaped structure on 25 June in areas where |V geo | is significantly larger than zero, indicating that the roll-up is a highly nonlinear process.

Vorticity and strain
The formation of the spiral can be considered as the deformation of an incompressible fluid, thus it is controlled by rotation 10 and shear strain, which are represented in the present case by the relative vorticity and the horizontal strain rate Here, x and y are the Cartesian coordinates pointing to the east and to the north, z is the vertical coordinate, and u, v are the 15 zonal and meridional components of the total velocity vector. Furthermore, the Okubo-Weiss parameter is a relative measure indicating whether the deformation is primarily driven by relative vorticity (η < 0) or horizontal strain (η > 0).
Near the sea surface, the normalized relative vorticity ζ = ζ/f (f is the Coriolis frequency) exhibits a maximum value of 20 12.3 during the coiling of the spiral on 25 June (Fig. 6b), while in the days before and after, the maximum is around 4. At the same time, the normalized anticyclonic vorticity reaches a minimum of -2.2 in an extremely narrow ribbon right to the south of the concave part of the cyclonic pattern. On the other days, it remains between -0.9 and 0. While in adiabatic flow, ζ > −1 should hold, ζ < −1 is probably a consequence of the diabatic contribution from the biharmonic diffusivity, caused by the extreme horizontal density gradient (cf. Fig. 4g). For comparison, we recall the observations of Marmorino et al. (2018). They 25 found maximum ζ max = 12.5 in the core of a cyclone comparable in size, that is remarkably close to our value. The f -scaled strain rate = /f is also maximum on 25 June (panel g) with values > 18, while on the other days it attains values between 4 and 6. Striking is the quadrupole shape that begins to evolve on 26 June and becomes clearly visible during the successive days. Another eye-catching feature is the spot with = 5.4 close to the center of the image in panel j. Also these numbers resemble closely the corresponding ones of Marmorino et al. (2018), max = 18.8 and a mean of 6.75. The patchy pattern of 30 in their Fig. 4F shows some agreement as well.  The scaled Okubo-Weiss parameter η = η/f 2 exhibits the largest spread on 25 June with −106 < η < 258 (panel l), indicating a bimodal structure with extremely strong control of both vorticity and even more strain. Here, vorticity dominates in the center of the spiral and along the hook. Maximum strain dominated areas are also constrained to the hook, but they are absent in the center. The day before (panel k), η reveals distinct positive areas, hence the northward progression of the filament is mainly controlled by deformation due to horizontal strain. However, the spread of η between about -3 and 17 is much smaller

Vertical motion
As ROMS is a hydrostatic model, the vertical velocity w(z) is computed from the horizontal divergence δ = ∂u/∂x + ∂v/∂y by integration, where H is the water depth. In ROMS, the integral is actually computed as a sum from the bottom upwards and also as a sum 15 from the top downwards, resulting in a linear combination of the two, weighted so that the surface down value is used near the surface while the other is used near the bottom (Hedström, 2018). Thus, the near-surface vertical velocity largely reflects the divergence pattern in the surface layer. This is shown in Fig day later, the convergence zone becomes strained and attains the hook-like shape mentioned above. To the south of the hook appears a large area of divergent flow and associated upwelling (panels b, g, l).
A perfect multi-arm spiral has finally developed on 26 June (panel c). Later on, the signs are mounting that the arms of the spiral disaggregate into less organized structures although the vortex is still intact. This tendency is confirmed by panels e and j; namely, the structure of the divergence looks still spiral-like, but the vertical motion exhibits an increasingly patchy 25 pattern (panels j, o). Concurrently, the spread of the vertical velocity at 5-m depth is increasing. While on 24 June, -45 m day −1 < w < 49 m day −1 , the spread on 26 and 27 June is -81 m day −1 < w < 60m day −1 .
The progressive decorrelation between the surface divergence and w at 5-m depth is due to sign changes of δ with depth according to equation (

Vortex Rossby waves
The above investigations have shown that the fully developed cyclone is not axisymmetric. While the density pattern (e.g. Fig.   4d) is ellipsoidal, many quantities are spiraliform, and other quantities such as the horizontal velocity, MLD, and h ρ exhibit 5 azimuthally irregular or wavy patterns. Similar non-axisymmetric structures, e.g. spiral-shaped rain or cloud bands, are also observed in atmospheric cyclones. According to various theoretical studies (Montgomery and Kallenbach, 1997;Chen and Yau, 2001;Wang, 2002a, b), they are caused by vortex Rossby waves (VRWs). Unlike planetary Rossby waves, the restoring force of which is the meridional gradient of the Coriolis frequency, VRWs owe their existence to the radial and azimuthal gradients of the absolute vorticity. While the radial gradient enables their outward propagation, the orbital progression is driven by the 10 azimuthal gradient, and the common action of both gradients generates the spiraliform patterns.
In the ocean, VRWs were reported for the first time by Buongiorno Nardelli (2013)  In agreement with the papers of Viúdez and Dritschel (2004) and Pallàs Sanz and Viúdez (2005), the horizontal advection of relative vorticity was found to be the main driver for the vertical motion described in both publications.
In the present case, VRWs attracted our attention due to the wavy h ρ pattern in Fig. 8h, i, j. The most striking example was found on 26 June at 22h, where the radial excursions of the h ρ = 5-m contour attained a maximum of about 300 m (Fig. 9a).
The wavelength of the oscillations on all h ρ -contours is on the order of 100 m, and from the animation of Onken et al. (2021a), 20 a rather crude estimate of the cyclonic angular velocity arrived at 4 × 10 −5 rad s −1 , equivalent to an orbital period around 40 hours. According to the findings above, the vorticity advection ADV = V · ∇ζ a is the main driver of the vertical motion in eddies, where ζ a = ζ + f is the absolute vorticity. The components of ADV at 2-m depth are displayed in panels b and c. The azimuthal oscillations of the h ρ -contours are reflected by the horizontal speed |V | in panel b. Thus, water particles circulating the cyclone accelerate and decelerate alternately. In panel c, ζ a /f exhibits a rather complex pattern. The initial spiral (cf. Fig.   25 6) is still discernable, but in the eddy center emerged anomalous structures. The ADV structure (panel d) resembles almost perfectly the top-layer divergence and the corresponding vertical motion patterns in Fig. 7d and i, which is two hours later.
Instead of ADV , we have plotted −ADV in order to match the colors. Hence, positive vorticity advection (blue) correlates with convergent flow and downwelling, while ADV < 0 (red) indicates divergence and upwelling.
The most important message of Fig. 9 is that the spiral structure of the vertical motion pattern is caused by VRWs. If there 30 were no VRWs, the eddy would be circular and no non-zero vorticity transport would be accomplished, because the vectors of the horizontal velocity were parallel to contours of ζ a . Such a situation is approximately met in the southwest quadrant of panel c. Consequently, the spiral arms there are less pronounced.

Secondary instabilities
The objective for the setup of R33 was to provide a detailed description of a cyclonic spiral that was generated in the coarser resolution parent model R100. As the diameter of the spiral was about 1 km and no smaller spirals were found in R100, it was speculated that the minimum size of circular eddies in any circulation model was about 10 times the grid size. This is in agreement with the grandparent model R500 which did not generate any eddies smaller than about 5 km (Onken et al.,5 2020a). Consequently if this rule holds, R33 should be able to create eddies even smaller than 1 km. And in fact, several of such mini eddies were detected in the R33 model run. The most prominent example is exhibited in Fig (2016)). Moreover, in the framework of the 15 "Expedition Clockwork Ocean" (https://uhrwerk-ozean.de/, last access 10 November 2020), the evolution of an even smaller eddy of about 300 m in diameter was observed in the Baltic Sea (Fig. 11).

Drifters and floats
As atmospheric forcing is turned off, the dynamics of R33 can be considered as adiabatic and nearly frictionless because no explicit vertical mixing is specified, and the biharmonic diffusivity coefficient is extremely small. Hence, for any material water 20 particle, is the potential vorticity expressed in isopycnal coordinates (Müller, 1995). Here, ρ 0 is the mean density of the respective 25 isopycnal layer, ζ a ρ = ζ ρ + f the absolute vorticity, and ζ ρ the relative vorticity evaluated along a surface of constant density. H is the vertical spacing between the shallower isopycnal with density ρ 0 − ∆ρ/2 and the deeper one with density ρ 0 + ∆ρ/2, where ∆ρ is the density difference between the deeper and the shallower isopycnal. Because the R33 domain is rather small with about 20 × 20 km 2 , f is considered to be constant. Hence, material changes of the absolute vorticity are compensated by changes of the layer thickness and vice versa: For instance, the height of a water column increases, if the vorticity increases (vortex stretching), and a decrease of the vorticity causes shrinking of the water column (vortex squeezing). In the northern hemisphere and in the absence of diabatic processes, q is always positive (Harvey, 2020), i.e. ζ a ρ >= 0. Consequently, anticyclonic relative vorticity cannot be smaller than the negative Coriolis parameter while the values of cyclonic vorticity are not constrained. An important consequence of potential vorticity conservation is that material changes of any kinematic or dynamical quantity are intimately and mutually 5 related to material changes of ζ a ρ or H. For example, convergent flow (δ < 0) is a consequence of vortex stretching due to increasing vorticity or vice-versa, while δ > 0 is correlated with decreasing vorticity. In the following and in order to explore and understand some of its properties, the cyclonic spiral is larded with isopycnal floats which are assumed to represent material (or Lagrangian) water particles conserving potential vorticity. Before, however, two precursor experiments are conducted with isobaric floats that are launched close to the sea surface. Namely, such floats do not conserve potential vorticity but it is expected that they behave like real surface drifters; therefore, they are denoted as isobaric drifters or just "drifters".

Isobaric drifters
In the first experiment, named ISOBAR1 (see Table 1  The (ambient) density of the 48 drifters over the course of the model run is shown in Fig. 13a in a contour plot. The initial density at 0 hours reflects the above description with just one density maximum for #54 and #55, and minima for #51 and #98.
At the end of the model run at 96 hours, the arrangement has changed: now, there are two density maxima centered at about #57 and #85, two minima for #51 an #98, and a third minimum centered at about #70. That means that some drifters or drifter clusters swapt their position relative to the absolute density maximum. By way of example, we compare the histories of #70 5 and #90: as at 0 hours, the density of #70 was higher than the density of #90, #70 was closer to the high-density center of the filament. In contrast at 96 hours, the density of #90 is higher than that of #70, thus #90 is closer to the center of the spiral. This The above experiment included only 134 drifters -it was therefore easy to identify the ones that were captured by the circular 30 circulation of the eddy and to describe the individual properties of the ambient fluid along their track. However, the quantity of drifters was not sufficient in order to simulate any observed tracer pattern at the sea surface. Therefore, we repeated the experiment with 100,000 drifters, that were randomly distributed in a rectangular area on 24 June (Fig. 14b). This experiment is denoted ISOBAR2 (see again Table 1) . Two days later on later on 26 June (Fig. 14a)  single-arm shapes of the surface density front and the vorticity, but they are different from the pattern of the multi-arm shaped horizontal divergence shown in Fig. 7c. In conclusion, the spiral structure of the cyanobacteria in Fig. 1 was certainly generated by alternating convergent and divergent currents but it may not reflect the instantaneous divergence.
On may argue that the behavior of the drifters is unrealistic because the density is modified along their path. Indeed, this must not happen with Lagrangian floats representing material water particles, but as isobaric drifters are forced to stay at a 5 predefined pressure level, they cannot behave like material particles. For instance, in convergent flow, density is merely an Eulerian environmental parameter for isobaric drifters that changes in response to the subduction process, while Lagrangian (isopycnal) floats subside in order to preserve their material property "density". Amongst others, this different behavior is demonstrated in the following subsection.

Isopycnal floats 10
An additional experiment is carried out with 134 isopycnal floats (ISOPYC1, see Table 1). The floats are launched exactly at the same positions and at the same time as in the experiment ISOBAR1 with the isobaric drifters. The horizontal positions of the floats are displayed in Fig. 15. After 12 hours (panel b), they closely resemble the corresponding positions of the isobaric drifters (cf. Fig. 12b), but already on 25 June (panels c), the floats trapped by the cyclonic spiral cover a larger area than the corresponding isobaric drifters. This comparison is legitimate because the same floats #51 -#98 plus #99 and #100 participate 15 in the early stage of the coiling, using the same numbering as in the experiment ISOBAR1. However, the behavior of #53 -#55 (black dots) is rather remarkable: instead of performing orbital motions around the center of the spiral, they veer immediately away from the cyclone into the ambient water and continue traveling in southwesterly direction (panels c-f). Surprisingly, the initial positions of these runaways were located inside the density maximum of the filament (panel a) that became the center of the spiral. Later on during the course of the integration, the remaining floats stay inside the cyclone.

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Time series of the density of floats #51 -#100 (Fig. 16a) reveal that this quantity is reasonably conserved for all floats during the entire model run, except for #53, the final density of which on 28 June is 0.09 kg m −3 lower than the initial density. During the first 48 hours, all floats sink to greater depth (Fig. 16b). Thereafter, many of them raise again towards the surface, but others stay at depth or oscillate vertically. By consideration of the maximum depths, the floats may be separated into 3 groups: -Group I floats (magenta dots in Fig. 15) descend to maximum depths of about 5 m; the 24 members of this group are #58

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-#74 and #94 -#100. According to Fig. 15a, #58 -#74 originate from the inner part of the filament and descend to a maximum depth of 4-5 m between about hours 12 and 24; most of them return to shallower levels thereafter. In contrast, the initial positions of #94 -#100 are about 2 km to the east of the center of the filament; they descend almost constantly to their final depth between 3 m and 4 m.
-Group II floats (orange) descend to maximum depths between 5 and 10 m; the 23 member of this group are #51, #52, 30 #56, #57, and #75 -#93; their initial positions were located very close to the the density maximum of the filament and between about 1 and 2 km east of the maximum.  . This low-frequency vertical oscillation is apparently a consequence of the subduction of the corresponding isopycnals as illustrated by Fig. 8f, g, k, l, and a subsequent equilibration.
Later than hour 30, the float performs yo-yo like high-frequency vertical oscillations with small amplitude according to panel c. We postulate that these oscillations are driven by material changes of the vorticity that is compensated by changes of the 10 layer thickness according to equation (8). The latter in turn requires non-zero divergence and associated vertical velocities as specified by equation (5). In order to demonstrate the interplay of these quantities, we performed an OSSE-like (Halliwell et al., 2017) simulation experiment, where the float trajectory stands for the observations and the ROMS output represents the environment of the float, i.e. the truth. As the float is isopycnal, we projected in the first step the prognostic variables of the ROMS output on the 5.243 kg m −3 isopycnal and computed ζ a ρ , H, and q using ∆ρ = 0.01 kg m −3 (see equations (7) and 15 (8)). The result is displayed in Fig. 17. The time series in panels a, b, c are created from the trajectory data which are available at each model time step of 20 s. In contrast, the graphs in panels d, e, f are extracted from the ROMS output. As the output is only available in intervals of 6 minutes, ζ a ρ , H, and q were interpolated in time and along the respective isopycnal in space at the exact horizontal position of the float. Fig. 17b shows the vertical position of the float. Altough this plot contains the same information as Fig. 16b, it exhibits high-frequency oscillations that are merely visible in the contour plot. These oscillations 20 are even more pronounced in the graph of the vertical velocity w (panel c), which was computed from the time derivative of z. Rapid oscillations are also characteristic for ζ a ρ and H (panel d, e) for times later than 27 hours. Before, there is no clear signal detectable, because the isopycnal ρ 0 outcrops intermittently at the sea surface. The critical question arises whether ζ a ρ and H are related to each other. And indeed, a correlation analysis yields a significant coefficient of 0.64; hence, fractional changes of the absolute vorticity equal largely fractional changes of the layer thickness as required by equation (8). Ideally, q 25 should be individually conserved, but according to panel f, it fluctuates between about 0.6 × 10 −3 m −1 s −1 and 1.3 × 10 −3 m −1 s −1 . Moreover, ζ a ρ and w are not at all correlated, in agreement with Buongiorno Nardelli (2013). Namely, there is a direct relationship between ζ a ρ , H and the divergence δ, however, w does not depend on δ but on its vertical integral (equation (5)). What has been said above implies that material vorticity changes drive the vertical motion. By way of example, Fig. 18 illustrates how such changes may come about. It shows the positions of float #64 at four different instants in one-hour intervals 30 on 26 June, together with a map of the absolute isopycnal vorticity. At 9:00 h (panel a), the vorticity at the position of the float is around 3, at 10:00 h it is greater than 4, and thereafter it decreases again to about 3 at 11:00 h and less than 3 at noon. This sequence is reflected by the gray-shaded patch in Fig. 17d. Hence, the material vorticity changes are due to intersections of the float track with vorticity contours. As the float track is a proxy for the horizontal velocity, this is equivalent to non-zero vorticity advection, the same mechanism that drives VRWs (cf. Figs. 6c, d and Fig. 9).    For further illustration of the kinematics during the roll-up process of the spiral, nine cutouts of an animation of the float tracks together with the shape of an isopycnal surface (Onken et al., 2021b) are lumped together in Fig. 19. The isopycnal ρ = 25.3 kg m −3 is shown for two reasons: (i) this surface outcrops at the sea surface at the start of the integration on 24 June and subducts thereafter. The subduction process is finished on 24 June at 18:00 and the isopycnal remains at a depth of about 3-4 m in the center of the spiral (Fig. 8, center and right column). (ii) The majority of floats that get caught by the spiral (except 5 for #51 -#57, cf. Fig. 16a) remain above that isopycnal during the first 36 hours. In the morning of 24 June, the isopycnal is shaped like a slightly curved mountain ridge. The curvature radius decreases rapidly and a spiraliform ridge starts to form in the evening of the same day. The spiral is clearly visible on 25 June and explains the generation mechanism of the wavelike patterns on the flanks of the isopycnal in Fig. 8. Later on, the height of the ridge flattens out, but groovings around the doming isopycnal in the center of the eddy are still visible. The groovings are VRWs, and they circulate slowly anticlockwise around 10 the central dome which much smaller angular speed than the angular speed of the floats. As the floats are isopycnal, they have to respond to the changing depth of the isopycnal along their path by vertical motion. This is in agreement with Buongiorno Nardelli (2013) and clearly visible in the above mentioned animation and the wavelike tails of the floats in Fig. 19.
The above experiment has shown that the dynamics of an evolving spiral leads to an export of Lagrangian particles from the near-surface layers to greater depths. Apparently, the vertical displacement of the particles depends on the initial radial 15 distance from the density maximum of the filament, but it is not a monotonic function of the distance, because both Group I and Group II particles are organized each in 2 modes, where the first mode is closer to the density maximum and the second is farther away. The modes of these groups are alternately arranged, and it is conjectured that the associated vertical displacements in the spiral exhibit a banded circular structure. However, as this is difficult to demonstrate with just 50 floats, we repeated the experiment with 100,000 floats in order to obtain a statistically robust outcome (ISOPYC2, see Table 1). In Fig. 20a, the 20 horizontal positions of floats on 26 June are marked by dots. Their color code indicates the attained depths at this particular time. Similar to the experiment with 100,000 isobaric drifters (cf. Fig. 14), the single-arm spiral of the cyclone is visible by means of the elevated distribution density of floats. While the dark and bright blue colors in the center of the spiral and adjacent to the wrapped arm suggest attained depths in the 0-5-m range, the light blue and green dots in the arm denote greater depth of up to about 12 m. This is an indication for enhanced downwelling in the spiral arm and a radially banded structure of vertical 25 motions. Fig. 20a also provides clear evidence for the existence of 287 runaway floats that subducted in the center of the spiral and thereafter escaped from the orbiting motion in the thermocline. The runaways are highlighted by green to red colors and attain maximum depths of up to 18 m. Fig. 20b proves that these floats started in the center of the filament on 24 June, because their mean density at the deployment depth was 5.304 ± 0.010 kg m −3 (panel c). At that time, such high densities were only present in the filament and north of 54 • 42'N, but not any float was deployed in the latter region (see Fig. 14a).

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In order to explore whether there is also isopycnal transport in the opposite direction (upwelling), ISOPYC2 was repeated, but now, the initial depths of the floats were set to 10 m below the sea surface instead of 1 m (ISOPYC3, see Table 1). In   Fig. 18). The full animation is shown in Onken et al. (2021b). rose by a few meters. According to panel b, most of them were entrained from the region west of the filament, but a significant amount was also absorbed by the coiling filament while proceeding northward. The majority of sinking floats is found west of 14 • 39' E. These floats originate from the center of the filament, similar to the runaways in ISOPYC2. Hence, there exists apparently a mechanism close to the center of the evolving spiral that procures an efficient export of particles to greater depth, and that finally enables them to escape from the vortex. The objective of the final experiment, ISOPYC4 (see Table 1), was to 5 determine whether that mechanism is a permanent feature of the spiral. Therefore, 100,000 floats were deployed 1 day later than in all previous experiments on 25 June, again at 1-m depth and at the same positions as in ISOPYC2. Namely, there were the intense downwelling is restricted to the early formation phase of the spiral.

Dispersion
We define the dispersion of a cluster of N floats as

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The dispersion curve of the 50 isopycnal floats clustered by the flow in the spiral resembles the corresponding curve of the drifters, but the values are generally higher. The reason for this is that the isopycnal floats have to sink in response to surface convergence, and the sinking can only occur away from the center along the sloping isopycnals. The decreasing dispersion between hours 20 and 33 is not in contradiction with that, because the majority of the Group I and Group II floats returns quite rapidly from their maximum depth to shallower levels, getting closer to the center. Moreover, instead of a further decrease, the dispersion of the shallow floats starts to increase again after 33 hours, which is most likely caused by Group I floats, sinking almost steadily during the first 3 days.

Model setup
The evolution of a submesoscale cyclonic spiral was simulated by means of a triple-nested ROMS setup. This method is, of course, rather complex and expensive, but it allows to explore the kinematics of the spiral from the very beginning of a filament until it reaches a mature stage. Alternatively, one might have started from an idealized circular-shaped eddy like McGillicuddy (2015), but that would not reproduce the asymmetries of the coiling filament and the spiraliform vorticity. The latter, in turn, 10 enforces non-zero vorticity advection and the associated VRWs, that finally shape the vertical motion pattern.
Atmospheric forcing, i.e. the surface fluxes of momentum, heat and freshwater, were turned off in R33 in accordance with the parent model R100. Realistic fluxes were only applied in the grandparent R500 in order to generate a mesoscale environment that is representative for early summer in the southern Baltic Sea. The turning-off is justified because mainly the momentum flux impedes the generation of submesoscale instabilities (Renault et al., 2018;Kubryakov and Stanichny, 2015;Zatsepin et al., 2005) and blurs the corresponding structures as shown by Onken et al. (2020a) and Mahadevan et al. (2010). Different processes may lead to the generation of spirals, e.g. winding of filaments around existing vortices (Zhang and McGillicuddy, 2020), lateral straining of tracer fields (Meunier et al., 2019), or barotropic/baroclinic instability (de Marez et al., 2020). All these processes are adiabatic, or at least, diabatic forcing is not required to get them going. Moreover, isopycnic potential 5 vorticity is conserved under these condition, which facilitates the interpretation of the model results.
The Lagrangian floats in ROMS follow neutral surfaces (van Sebille et al., 2018), but our analyses of the individual properties of the floats are based on the assumption that the floats move along potential density surfaces. This is acceptable, because the differences between neutral surfaces and potential density surfaces are negligible, as long as the difference between the reference pressure and the in situ pressure is small (You and McDougall, 1990). The latter applies for all floats. Nevertheless, 10 as shown above (e.g. Figs. 16a, 20c), the density of several floats changes during the course of the integration, and concurrently the isopycnal potential vorticity. Thus, one is tempted to assume that ROMS does not conserve potential vorticity, but that would be a very strong statement and call the validity of the primitive equations or their finite difference equivalents into question. Instead, it is more likely that isopycnic potential vorticity is not conserved in a non-isopycnic coordinate system.
Other conceivable reasons could be the even though small diffusivity or that the float tracks are inaccurately computed, which 15 makes them different from the paths of actual Lagrangian water particles.

Impact of downscaling
For the initialization of R33 on 24 June, the corresponding prognostic fields of R100 were mapped onto the R33 grid by linear interpolation. As the s-coordinates of the parent (R100) and the child (R33) were identical, the vertical interpolation did not imply any approximations. In contrast, the mapping from the coarse-resolution horizontal parent grid on the higher-resolution 20 grid of the child is a potential source of error, because the interpolation may generate local imbalances of the involved forces.
Furthermore during the integration, boundary values were provided by R100 in 3-hour intervals along the open boundaries. As the baroclinic time step in R33 was 20 s, all prognostic variables along the open boundaries were updated by linear interpolation at each time step, which may create additional errors. In order to minimize such errors, it is recommended that the grid refinement factor should not be much larger than 3 (McWilliams, 2016), which is the case in our setup. Nevertheless, the 25 integration of the primitive equations may produce a solution in the child domain that is consistent with the initial and boundary conditions but deviates significantly from the solution of the parent. A tendency for R33 to develop its own solution, that is different from R100, is discernable in Fig. 3 already on 26 June, merely 2 days after the initialization. However, this does not necessarily imply that the solution is erroneous, particularly because there are no visible primary errors of downscaling procedures, so-called rim currents (Mason et al., 2010) along the open boundaries. Moreover, the evolving spiral as the main 30 object of our research is apparently not affected.
Despite of the potential shortcomings, the advantages of downscaling are striking, because the high-resolution child resolves features that are not visible in the solution of the parent. This is demonstrated by means of selected quantities in Fig. 23: the spiraliform shape of the surface density front in R33 is barely visible in R100 (cf. panels a, f); in the same way, R100 pretends a circular structure of the relative vorticity (panel c), that is a spiral as well ( Table 2, we have collated the extreme values of various near-surface quantities and computed the ratio of the extremes between the child and the parent. A 5 large ratio of 5.8 was obtained for the frontogenetic tendency, being 64.0 × 10 −12 kg −2 m −8 s −1 in R33 and just 11.1 × 10 −12 kg −2 m −8 s −1 in R100. Hence, the child is almost 6 times more frontogenetic than the parent, which is also reflected by the 6.3 times higher ratio of the absolute horizontal density gradient. Similarly, this applies for the frontolytic tendency, where the ratio is 4.1. Large ratios of 4.9 (upwelling) and 6.3 (downwelling) also predominate for the vertical velocity, the normalized horizontal divergence with ratios of 6.0 and 3.0 for convergent and divergent flow, and the normalized strain rate with 6.5. In 10 contrast, smaller ratios between 3 and 4 are found for the normalized relative vorticity, while for the ageostrophic speed the ratio is even less at 2.1. Finally, the maxima of the total and the geostrophic speeds reach about the same magnitude. Thus, in addition to the higher resolution of various features, the downscaling enhances particularly the amplification and attenuation of fronts and the intensification of the vertical velocity field.

Properties of the spiral 15
The properties of the spiral were already compared with previous studies above. As there were no numerical models known to the authors focusing explicitly on the properties of submesoscale eddies and least of all of cyclonic spirals, the model results were checked against just 2 observational studies, namely those of Ohlmann et al. (2017) and Marmorino et al. (2018). These resembled astonishingly well the modeled patterns of relative vorticity, strain rate, and the radial structure of the horizontal results from other models, and the formation of secondary instabilities was validated by our own observations (Fig. 11). These agreements support the results of the present study.

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The sketch in Fig. 24 summarizes our findings with respect to the vertical circulation during the formation phase of the spiral. In the near-surface layers, downwelling and corresponding subduction of isopycnals is invoked by the overall convergent flow as suggested by the decreasing dispersion of the drifters captured by the spiral (Fig. 22). The mean vertical velocity was estimated from the vertical displacements of isopycnal floats during the first 2 days. It exhibits radially arranged bands with enhanced downwelling, that are congruent with the spiral of elevated cyclonic vorticity; hence, the downwelling is apparently forced by 15 material changes of the relative vorticity, which cause vortex stretching (Fig. 17) and associated convergent horizontal flow.
According to Figs. 16b, and 20a, maximum downwelling of up to 18 m day −1 prevails in the center of the spiral for the "shallow" floats that were deployed at 1-m depth; the "deep" floats deployed at 10-m depth descended at about half that speed ( Fig. 21a). Downward velocities of the shallow floats up to 10 m day −1 are also found in the bands with enhanced downwelling, while the downwelling between the bands is rather modest, reaching not more than about 5 m day −1 . In contrast and in order 20 to compensate for the subducted amount of water, the peripheral deep floats rise at a about the same speed, opposing the sinking shallow floats from above. However, it is not clear whether the shallow and the deep floats really converge in the same isopycnal layer or whether they glide on top of each other as suggested by Fig. 7m. One should also bear in mind that the circulation scheme in Fig. 24 is based on mean vertical displacements. In reality, the instantaneous vertical motion pattern in the spiral is organized more slab-like with neighboring up-and downwelling cells as shown in Fig. 7h. These slabs are created 25 by VRWs with a revolution period around the eddy center (estimated to 40 hours, see above Section 3.6) that is much larger than the orbital period of about 10 hours of a water particle at 500-m radius, assuming an azimuthal speed of 10 cm s −1 . Hence, each Lagrangian particle experiences high-frequency vertical oscillations along its trajectory as shown in Fig. 17. While both the shallow and the deep peripheral floats continue to orbit the vortex during their vertical displacement, those starting in the density maximum of the filament feature the same capability to escape immediately from the cyclonic circulation already 30 during the first day. According to the trajectories of the runaways, they slip rapidly down along isopycnals and thereafter tunnel under the swirling flow in the upper layers.
In a nutshell, downwelling prevails near the surface and in the center of the spiral at all depth levels. It is driven by convergent near-surface flow and associated subduction of isopycnals. Compensating weak upwelling is found above 10-m depth at  Bakun (2006) who distinguished between forced ("spinning up") and free ("spinning down") eddies in a two-layer model: in case of a forced cyclone, the frictional torque is positive. This effectuates an intensification of the cyclonic vorticity which in turn leads to upwelling in the eddy core and associated divergent flow at the sea surface. On the other hand, in a free cyclone, that is not trapped by topography or coasts, the frictional torque is negative, and the diminishing cyclonic vorticity forces downwelling in the core with convergent surface flow and an outward directed flow in the lower layer.
Analogously, the circulation is inverse in anticyclones as confirmed by Bashmachnikov et al. (2019).

5
In R33, the evolution of the spiral is tracked over 4 days, starting from the very early stage of a dense filament on 24 June.
During the first day, the filament is rolled into a vortex, and the roll-up is completed in the morning of 25 June when the density anomaly of the spiral separates from the filament. Subsequently, the mature spiral is tracked until 28 June. Based on these heuristic arguments, one may denote therefore the first about 30 hours as the spin-up phase of the spiral. However, the modeled spiral during this period of time is not a "spinning-up" cyclone with upwelling in the center as defined by Bakun 10 (2006). That type of cyclone requires external frictional forcing which is absent in the modeled cyclone. Instead, patches of up-and downwelling alternate due to the convergent spiral arms of w (Fig. 7), and the mean direction of the vertical velocity is downward. Hence from the beginning, the modeled spiral is a free cyclone, that exhibits the characteristics of a "spinningdown" cyclone. This is also consistent with the subduction of isopynals (cf. Figs. 8k-o and 19), that starts already in the early morning of 24 June, i.e. the isopycnals where never lifted up ("eddy pumping") as claimed by Flierl and McGillicuddy (2002) 15 and McGillicuddy (2016).
In the light of the findings above, one may reinterpret the only existing observational study of a submesoscale spiral, which appears to be in conflict with our results. That spiral was observed by Marmorino et al. (2018) in the Southern California Counter Current (SCCC) at the northern tip of Santa Catalina Island (see the "study area" in the inset of their Figure 1).
Assuming that both our results and the observations are not faulty, we are left to interpret the different circulation patterns in 20 terms of the schemes of Flierl and McGillicuddy (2002) and Bakun (2006). Accordingly, the modeled spiral is a free cyclone, while the secondary circulation of the observed eddy exhibits typical characteristics of a forced cyclone. As shown by the model experiments of Dong and McWilliams (2007), the forcing is due to lateral and bottom stresses, forming sheets of positive vorticity and associated current-wake instability (Marmorino et al., 2010) when the SCCC flows along the northeast coast of Santa Catalina Island. A part of the sheet leaves from the northern tip of the island and transforms into a circular 25 cyclonic spiral and continues to propagate downstream with the SCCC.

Implications
Our model results characterize the ageostrophic secondary circulation in an evolving submesoscale cyclonic eddy. In analogy to dense filaments McWilliams, 2017), the secondary circulation is an overturning cell with intense downwelling in the eddy center and weak upwelling at the periphery. Besides the ability to accumulate ice (Manucharyan However, the convergence is neither uniformly spread across the cyclone nor concentrated in a single point in the center; instead, it is organized in multi-arm spirals with negative helicity comprising alternating bands of convergent and divergent flow. As downwelling is intimately tied to convergent flow, it affects only particles with negative or zero buoyancy. According to Gemmel et al. (2016) and Lännergren (1979), the majority of phytoplankton species are negatively buoyant, although there are a few exemptions (Woods and Villareal, 2008). Hence during the plankton bloom, when the nutrients in the near-surface layers are depleted, phytoplankton cells at the periphery of the spiral may sink by a few meters within a weakly stratified water column and find more favorable conditions, particularly because of the upward transport of nutrients on the same isopycnals ( Fig. 24). Potentially, this mechanism contributes to the formation of chlorophyll rings in mesoscale eddies as observed by Xu et al. (2019). In contrast, cells in the center of the spiral reach depth levels between 15 m and 20 m, i.e. within the seasonal thermocline. This eddy-driven subduction may support the bloom of cyanobacteria at deeper levels in July and August (Hajdu et al., 2007) and beyond that it reinforces the export of particulate organic carbon from the mixed layer (Omand et al., 2015;10 von Appen et al., 2018).

Transferability
Details of the modeled spiral were partly compared with the corresponding properties in mesoscale eddies. This was necessary because of the lack of information with respect to the appropriate submesoscale features, both from observations and models.
As frequently sufficient agreement was found, this raises the question whether and to what extent our findings can be conversely 15 transferred to structures and motions in mesoscale eddies. To begin with mesoscale cyclonic spirals, it is unclear how often they occur. Namely, by means of SAR (synthetic aperture radar) images, Dokken and Wahl (1996), DiGiacomo and Holt (2001), Karimova and Gade (2016), and Karimova (2012) detected thousands of cyclonic spirals in various waters, but the diameters of the vast majority (partly more then 99%) was less than 20 km, suggesting that most of the spirals are submesoscale features.
However, that does not mean that mesoscale spirals do not exist; potentially, they are more difficult to discover with SAR 20 because the required changes of surface roughness are too weak on larger scales. In contrast, by combining ocean-color, satellite altimetry and surface drifter data, Zhang and Qiu (2020) demonstrated that spiral bands of chlorophyll enhancement emerge globally both in cyclonic and anticyclonic mesoscale eddies, and Zhang and McGillicuddy (2020) verified the existence of spiraliform streamers in Gulf Stream anticyclonic rings by means of satellite-measured sea surface temperature. Hence, both submesoscale and mesoscale spirals appear to be common features in the ocean. However, submesoscale spirals are cyclonic, 25 nearly without exception, and they develop from a filament which is rolled in a spiraliform vortex. By contrast, mesoscale spirals are both cyclonic and anticyclonic, and they are formed in rings that have been detached from unstable jets. Although the forms of appearance of all spirals are very similar or even identical, this does not necessarily mean that the dynamics are the same, because the balance of forces is different for submesoscale and mesoscale motions.

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The evolution of a small submesoscale cyclonic spiral is simulated by means of a high-resolution circulation model. The generation of the unforced spiral starts from a dense filament that is rolled into a vortex and becomes detached from the filament. The first about 36 hours of that process are referred to as the formation or spin-up, and the time after as the decay or spin-down of the spiral. Here, the meanings of "formation" and "spin-up" differ from the definitions of Flierl and McGillicuddy (2002) and Bakun (2006), who used the identical wordings for forced eddies.
The spin-up is accomplished by a pre-existing mesoscale circulation pattern, that transforms the straight filament into a spiral. During this phase, all-time extreme values are attained by various quantities. While some become organized in single-5 arm spirals, multi-arm spirals with alternating signs are characteristic patterns of divergence, frontal tendency, ageostrophic, and vertical velocity. The multi-arm spirals are forced by vortex Rossby waves, that are excited by advection of vorticity. The ageostrophic together with the vertical velocity effectuate the overall decrease of baroclinicity and contemporaneous subduction of isopycnals and mixed-layer shoaling, indicating baroclinic instability as the main driver for the formation of the spiral.
The spin-down starts when the cyclone separates from the filament. At the same time, the horizontal speed develops a dipole-10 like pattern and isotachs generate closed contours around the center of the vortex. The amplitude of most quantities decreases significantly, except for the instantaneous vertical velocity, that attains even more pronounced extremes than during spin-up.
Secondary instabilities in the wake of the spiral are potential gateways for the cascade of energy towards the microscale.
The mean ageostrophic secondary circulation comprises a circular symmetric overturning cell with intense downwelling in the center and weak upwelling at the periphery. The associated convergent flow at the surface may concentrate buoyant flotsam 15 in the spiral. The downwelling in the center reinforces the export of particulate organic carbon from the mixed layer into the main thermocline. The upward isopycnal transport of nutrients at the periphery supports the growth of phytoplankton in the euphotic zone.
Code and data availability. The model code and the output of the ROMS runs presented in this article are available from the first author on request.

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Author contributions. Reiner Onken has set up and analyzed the model studies and has written the manuscript. Burkard Baschek has interpreted the results in the context of submesoscale eddy dynamics and the connection to the "Expedition Clockwork Ocean" that motivated this study. He also edited the manuscript and provided Figures 11 and 24.
Competing interests. The authors declare that they have no conflicts of interest.
Acknowledgements. The present study was funded by Helmholtz-Zentrum Geesthacht/Hereon through the program of the Helmholtz Asso-