Canyon Vortices: Application of the Theory of Topographic Vortices to the Phenomenon of Ice Rings in Baikal

Abstract

A phenomenon of ice rings in Lake Baikal is considered. A hydrodynamic theory is proposed according to which geostrophic vortices form under ice in the area near the ice rings. These vortices are due to bed relief, namely, underwater canyons; therefore, they are referred to as canyon vortices. It is shown that a divergent ring vortex forms under ice in the ring domain at the crossing of a vertical cylindrical Stewartson layer on the lateral surface of the canyon geostrophic vortex and under-ice horizontal Ekman layer in the form of a horizontal vortex torus, and the melting of ice below in the form of a ring is due to the formation of this vortex torus.

INTRODUCTION

Satellite images of 2009 for the first time showed gigantic rings in the ice cover of Baikal [2–4, 9, 11–14]. After that, such rings were regularly observed in satellite images. Detail studies of space images over previous years showed such rings to have formed even earlier than that, since the 1970s [13]. Similar rings were observed in different years since 1975 in Lake Khubsugul in Mongolia and Lake Teletskoe in the Altai [13, 14]. The rings are practically circular with a diameter of 5–7 km and a width of dark and thinner ice of ~1 km. It is obvious that the formations of such gigantic size can be seen only in satellite images with a sufficiently high resolution. The ice in the ring is thin (20–30 cm and less), which is of hazard for people and cars, while within and beyond the ring, ice is white and its thickness reaches 1 m (Fig. 1).

Fig. 1.

Ice rings in lakes Baikal and Khubsugul. Data from (a, c, g–i) Landsat 1–7, (d–f) MODIS/Terra, and (b) ISS. All images, except for (f) are oriented to N (from [13]).

The explanations of the formation of ice rings, proposed in the literature, are very diverse: from the effects of the atmosphere and biological processes in the top water layer to the effects of UFO, i.e., obvious ufologic mystifications, as for example, circles in fields etc. The most popular recent hypothesis is the hydrothermal activity of Lake Baikal, associated with the extremely high rate of gas (methane) emissions from bottom sediments in some places in the lake [2–4, 7, 11, 12]. In shallow zones, methane commonly enters water mass through its local sources and mud volcanos near faults on the bed, resulting in the formation of proparinas (small areas of open water from a few meters to tens of meters in size) in ice cover and the so-called sopkas. At greater depths, methane can release mostly from gas hydrates lying at depths >500 m in the southern and central depressions of Lake Baikal [8].

The mechanism of ring formation after gas release is still not quite clear. According to a hypothesis [2, 3], based on data of field observation from ice in April–May 2009 near the ring in the southern part of Lake Baikal at Slyudanka Town (Fig. 1b), as well as numerical simulation, the local upwelling after gas emission from a point source leads to the formation of a dome-like structure in water density field under the effect of Coriolis force and the formation of an anticyclonic (clockwise) geostrophic ring current. However, this hypothesis has weak points. One is due to the law of the extension of a vertical jet rising from the bed. D. Fox gave a solution to the problem of vertical jet in stably stratified fluid without external flow [16]. As follows from the proposed theory, the maximal angle of the cone of jet extension during its rise is 43°. The angle estimated by experiments is somewhat lesser, and the dependence of jet diameter d at distance z from the source of the liquid is described by the formula d = 0.68z. Taking into account that the depth of Baikal near the ring at Slyudyanka Town is 750 m, we find that the diameter of the ring on the surface cannot be greater than 550 m, i.e., the diameter of 4 km is absolutely impossible.

Other hypotheses, related to gas releases, incorporate the formation of toroidal convection caused by gas rise and delivering warmer water to the surface [7], or the formation of a gigantic has bubble under ice cover, which isolates the central part while increasing heat exchange at the outer boundary [11, 12]. Bordonskii et al. [1] attribute the formation of ice rings to deformations and heterogeneity of ice cover, as well as self-organization in the process of ice destruction in the form of a ring structure under the effect of incoming solar energy.

Kouraev et al. [13] present archives of satellite images and photographs and a complete list of the observed ice rings along with their parameters. Field hydrological studies conducted in 2010–2015 and 2019 showed that an anticyclonic vortex lies under the ring in the geostrophic domain of the lake. This vortex brings warmer deep water toward the bottom ice surface, leading to ice melting from below. A lens-like zone of mixed water forms under ice in the temperature field (Fig. 2) [14]. This suggested Kouraev et al. [13–15] an idea that the anticyclonic vortex under the ring is a within-thermocline lens. However, this hypothesis contains many considerable contradictions.

Fig. 2.

Vertical distribution of temperature in water mass under the ring in (a) February and (b) March 2016 (Baikal, Nizhnee Izgolov’e C.). The vertical lines show the location of stations [14].

The first observation that is difficult to explain is the velocity field in the vortex. According to [10], to which the authors of the above studies refer, the velocity vector in the domain immediately under the ring is to be first directed at some angle with respect to the tangent, but into the ring, next, further below, conversely, out of the ring, and even further, again into the ring. However, measurements show nothing of this kind (Fig. 3); the velocity vector is always directed out of the ring, i.e., the vortex carries liquid outside from the ring; therefore, the vortex has a divergent character.

Fig. 3.

Measured flow velocities under ice in sections via the zone of ice ring near Nizhnee Izgolov’e C. (Baikal, April 3, 2014) [13].

The second problem related to the lens is its permanent location under the ring. It is well known that a lens cannot stay without motion, it shows beta-drift, i.e., it has to move. However, in the case of ice rings, the lens stays under ice at the same place for months. The argument that the lens is captured by bed topography is also beneath criticism. The latter is possible if a compensation vortex forms under the lens in the bottom layer down to the bed, i.e., a heton forms. However, in this case, the bottom vortex is always a cyclone and the rise of deep water near the ring has no explanation.

It should be mentioned that such lens-like structure in the temperature field was first found by N.G. Granin et al. [4] in April 2009 under a ring near Slyudyanka (Fig. 1b) in the southern part of Lake Baikal. Later measurements, including those in February 2019 under the ring at Nizhnee Izgolov’e Cape (Fig. 2), once again confirmed such structure under the ring in temperature field. However, the most unclear in the hypothesis about within-thermocline lens is that it does not explain why ice melts in a ring zone. If the vortex has the form of a lens, its kinematics is as follows: water rises along the inner surface of the lens toward its center, and this means that ice will first melt in the center, i.e., an ice-hole should form. Note that the velocities of under-ice flows in the zone of convective mixing have a considerable effect on ice melting from below, i.e., the greater flow velocities, the faster ice melting. This idea could be applied to a lens, but in that case, the maximum of flow velocity would lie at a depth of 40–50 m, which is rather far from the ice surface. This means that lenses can hardly be here.

Thus, neither of the known hypotheses regarding the formation of ice rings can give an answer to the main question: why does ice melt from below along the ring? This article proposes an alternative theory of this phenomenon.

PROBLEM FORMULATION

Detailed analysis of the map of Baikal bed relief with the places where the rings appeared on it reveals an interesting coincidence―the rings commonly form in the corners of underwater canyons on lake bed [13]. The rings found in Lake Khubsugul in Mongolia also lie in the beginning/end of underwater canyons. The fact that the vortices under rings stay at the same place for months suggest that it is bed relief that holds them, i.e., they are directly tied to bathymetry. These facts indicate to the main formation factor of stationary anticyclonic vortices over the rings, i.e., the effect of bed relief, namely, the angular domains of underwater canyons. For some unclear reason, all researchers of ice rings ignore the basic factor in geophysical hydrodynamics: the architecture of stable meso- and large-scale currents is governed by bed relief. The positions of rings should be always superimposed on a bathymetric map.

The physics of the formation of canyon vortices is as follows: large-scale water motions in the geostrophic domain of the lake take place mostly along the contour lines of function f/H (f is Coriolis parameter; H is water depth). Because of the small variability of f, we can assume that the large-scale water motion along isobaths in Baikal depressions have cyclonic character, i.e., water moves counterclockwise. Consider a column of water between two isobaths near an angular point of a canyon. For definiteness, we assume that the canyon lies in the northern part of the depression, as, for example, the canyon at Nizhnee Izgolov’e Cape. Near the saddle point of the bed surface, the distance between the isobaths increases, as the bed topography becomes flatter. Figure 4 gives two types of underwater canyons for Baikal. The type (a) corresponds to the area near Nizhnee Izgolov’e Cape, where an ice ring forms, and the type (b) corresponds to the area of the Southern Baikal near Slyudyanka Town, where an ice ring also forms.

Fig. 4.

Schematic of two types of submarine canyons: (a) submarine canyon in a saddle point of bed relief; (b) canyon in an angular domain of shoreline (a bay).

In the initial state, water column A rotated with an angular velocity equal to that of the Earth at the given latitude; therefore, its momentum ${{L}_{z}}$ in the inertial coordinate system relative to the vertical axis will be ${{L}_{z}} = {{L}_{z}}{{\omega }}$ (${{L}_{z}}$ is the momentum of inertia of the liquid column, ${{\omega }}$ is the rotation speed of the liquid column; in this case, this is the projection of the Earth’s rotation velocity on the perpendicular to lake water surface). The momentum of inertia of a cylinder with radius $R$ is ${{J}_{z}} = {{m{{R}^{2}}} \mathord{\left/ {\vphantom {{m{{R}^{2}}} 2}} \right. \kern-0em} 2}$. The flow will carry water column A into position B (Figs. 4a, 4b). The isobaths diverge with approaching the angular point of the canyon; therefore, R will increase and, hence, ${{L}_{z}}$ will also increase. However, the angular momentum (impulse) must remain the same. Therefore, the rotation of the column at the angular point should slow down, which means that, in the coordinate system rigidly bound with the Earth, the rotation of the liquid column will become anticyclonic. Therefore, the situation under consideration is analogous to the formation of topographic vortices over underwater mountains. We can calculate an equivalent pseudo-mountain, which will generate the same vorticity.

Let us consider the situation in more detail. As can be seen from Fig. 4, initially, the circular liquid column А will take position В, and its horizontal cross-section will become elliptical. We denote the semi-major and semi-minor axes of the ellipse by ${{R}_{2}}$ and ${{R}_{3}}$, respectively, and the initial value of the radius of cylinder А, by ${{R}_{1}} = R$. Now, the equality of the volumes of the columns A and B, and, hence, their cross-section areas, yields ${{R}_{3}} = {{R_{1}^{2}} \mathord{\left/ {\vphantom {{R_{1}^{2}} {{{R}_{2}}}}} \right. \kern-0em} {{{R}_{2}}}}$. The momentum of inertia of an elliptic cylinder B relative to its vertical axis is ${{J}_{z}} = {{m(R_{2}^{2} + R_{3}^{2})} \mathord{\left/ {\vphantom {{m(R_{2}^{2} + R_{3}^{2})} 4}} \right. \kern-0em} 4}$. The law of conservation of momentum yields

$$\frac{m}{4}\left( {R_{2}^{2} + \frac{{R_{1}^{4}}}{{R_{2}^{2}}}} \right){{\omega }_{2}} = \frac{m}{2}R_{1}^{2}\omega .$$

(1а)

We introduce the denotation ${{\Delta \omega }} = {{{{\omega }}}_{2}} - {{\omega }}$. This is the additional vorticity that will manifest itself in reality (which will be observed). From (1a), we have

$$\Delta \omega = - \frac{{{{{(R_{2}^{2} - R_{1}^{2})}}^{2}}}}{{R_{1}^{4} + R_{2}^{4}}}\omega = - \bar {R}\omega .$$

(2а)

From (2a) we can see that water column B will acquire negative vorticity, i.e., it starts anticyclonic rotation. Note that we always have $\bar {R}$ < 1 in (2a).

Let us calculate an equivalent pseudomountain, which will produce the same vorticity. We have ${{\omega }} = {{{{\omega }}}_{E}}{\text{sin}}\phi$, ${{{{\omega }}}_{E}}$ is the angular velocity of Earth rotation, $\phi$ is the latitude of the site. Therefore, ${{\omega }} = {f \mathord{\left/ {\vphantom {f 2}} \right. \kern-0em} 2}$ (f is the Coriolis parameter), and (2a) can be written as ${{\Delta \omega }} = {{ - \bar {R}f} \mathord{\left/ {\vphantom {{ - \bar {R}f} 2}} \right. \kern-0em} 2}$. As the flow runs onto an underwater mountain with a height of h, the flow acquires anticyclonic vorticity, equal to ${{\Delta \omega }} = {{ - fh} \mathord{\left/ {\vphantom {{ - fh} H}} \right. \kern-0em} H}$ (H is water depth) [5]. Equating the obtained two expressions for $\Delta {{\omega }}$, we obtain the height of the equivalent pseudomountain, which produces exactly the same vortex:

$$h = {{\bar {R}H} \mathord{\left/ {\vphantom {{\bar {R}H} 2}} \right. \kern-0em} 2}.$$

(3а)

Now we can use the well-developed theory of topographic vortices [5] to obtain the kinematics of water flow in under-ice domain under the ring.

Ice melts in the under-ice layer adjacent to its bottom surface. In the rotating liquid, this is Ekman friction layer; therefore, the hydrodynamic under the ring is to be considered in the Ekman layer. Clearly, the vortex motion in Ekman layer under ice cannot appear spontaneously, it will be initiated by the vortex motion in the geostrophic domain, i.e., canyon vortex.

UNDER-ICE EKMAN LAYER

We consider an anticyclonic geostrophic vortex and The Ekman layer above it at the ice (Fig. 5). In the Ekman layer, the dimensionless complex horizontal velocity $U\left( {x,y,z} \right)$ takes the form [5]:

$$\begin{gathered} U = u + i{v} = - \frac{{\text{g}}}{{if}}\left( {\frac{{\partial {{P}_{{\text{g}}}}}}{{\partial x}} + i\frac{{\partial {{P}_{{\text{g}}}}}}{{\partial y}}} \right) \\ + \,\,{{C}_{1}}\exp \left( {\sqrt {\frac{{fi}}{{{{E}_{{v}}}}}} z} \right) + {{C}_{2}}\exp \left( { - \sqrt {\frac{{fi}}{{{{E}_{{v}}}}}} z} \right), \\ \end{gathered} $$

(1)

where i is imaginary unit; f is dimensionless Coriolis parameter; ${{E}_{{v}}} = {{{{A}_{z}}} \mathord{\left/ {\vphantom {{{{A}_{z}}} {{{f}_{0}}H_{0}^{2}}}} \right. \kern-0em} {{{f}_{0}}H_{0}^{2}}}$ is the vertical Ekman number; ${{A}_{z}}$ is the coefficient of vertical turbulent momentum exchange; g is the acceleration of gravity; ${{P}_{{\text{g}}}}$ is the geostrophic component of the pressure; ${{C}_{1}}$, ${{C}_{2}}$ are unknown constants; ${{H}_{0}}$ is depth; $z$ is dimensionless vertical coordinate.

Fig. 5.

To problem formulation: (1) geostrophic anticyclonic vortex; (2) Stewartson circular cylindrical layer; (3) under-ice Ekman layer; (4) bed, (5) ice.

Considering the no-slip conditions at the bottom boundary of ice and the limited velocity at infinity

$${{\left. U \right|}_{{z = 0}}} = 0,\,\,\,\,{{\left. U \right|}_{{z \to + \infty }}} < C < \infty ,$$

(2)

we obtain

$${{C}_{1}} = 0,\,\,\,\,{{C}_{2}} = \frac{{\text{g}}}{{if}}{{\left. {\left( {\frac{{\partial {{P}_{{\text{g}}}}}}{{\partial x}} + i\frac{{\partial {{P}_{{\text{g}}}}}}{{\partial y}}} \right)} \right|}_{{z = 0}}}.$$

(3)

We consider a homogeneous liquid. Now the gradient of ${{P}_{{\text{g}}}}$ does not depend on z, and we obtain the following expression for the velocity $U\left( {x,y,z} \right)$ in the Ekman layer:

$$U = u + i{v} = - \frac{{\text{g}}}{{if}}\left( {\frac{{\partial {{P}_{{\text{g}}}}}}{{\partial x}} + i\frac{{\partial {{P}_{{\text{g}}}}}}{{\partial y}}} \right)\left[ {1 - \exp \left( { - \sqrt {\frac{{fi}}{{{{E}_{{v}}}}}} z} \right)} \right].$$

(4)

We rewrite expression (4) for $U\left( {r,\phi ,z} \right)$ in cylindrical coordinates:

$$\begin{gathered} U = {{{v}}_{r}} + i{{{v}}_{\varphi }} = - \frac{{\text{g}}}{{if}}\left( {\frac{{\partial {{P}_{{\text{g}}}}}}{{\partial r}} + i\frac{{\partial {{P}_{{\text{g}}}}}}{{r\partial \varphi }}} \right) \\ \times \,\,\left[ {1 - \exp \left( { - \sqrt {\frac{{fi}}{{{{E}_{{v}}}}}} z} \right)} \right], \\ \end{gathered} $$

(5)

where ${{{v}}_{r}}\left( {r,z} \right)$ and ${{{v}}_{\phi }}\left( {r,z} \right)$ are the radial and azimuthal velocity components. From the axial symmetry of the geostrophic vortex, we have $\frac{{{{\delta }}{{P}_{{\text{g}}}}}}{{r{{\delta }}\phi }} = 0$. By virtue of geostrophics, we have

$${v}_{\varphi }^{{{\text{(g)}}}} = - \frac{{\text{g}}}{f}\frac{{\partial {{P}_{{\text{g}}}}}}{{\partial r}}.$$

(6)

Substituting (6) into (5) and separating the real and imaginary parts of the expression, we obtain the final expressions for the radial and azimuthal velocity components in the Ekman layer:

$$\begin{gathered} {{{v}}_{r}}(r,z) = {v}_{\varphi }^{{{\text{(g)}}}}(r)\exp ( - \alpha z)\sin (\alpha z), \\ {{{v}}_{\varphi }}(r,z) = - {v}_{\varphi }^{{{\text{(g)}}}}(r)[1 - \exp ( - \alpha z)\cos (\alpha z)], \\ \end{gathered} $$

(7)

where ${v}_{\phi }^{{\left( {\text{g}} \right)}}\left( r \right)$ is the azimuthal velocity in the geostrophic vortex, ${{\alpha }} = \sqrt {{f \mathord{\left/ {\vphantom {f {2{{E}_{{v}}}}}} \right. \kern-0em} {2{{E}_{{v}}}}}}$. The expressions for the flow velocity in the Ekman layer (8) contains only the azimuthal velocity in the geostrophic vortex.

Taking into account the above analogy between the formation of canyon and topographic vortices, we specify the structure of geostrophic vortex as topographic [5, 15].

For the azimuthal velocity ${v}_{\varphi }^{{\left( {\text{g}} \right)}}\left( r \right)$ in a geostrophic vortex with radius $R = 1$ we have the following expression [5, 15]:

$${v}_{\varphi }^{{{\text{(g)}}}}(r) = - \tilde {\sigma }\left\{ \begin{gathered} {{K}_{1}}(\sigma ){{I}_{1}}(\sigma r),\,\,\,\,r \leqslant 1 \hfill \\ {{I}_{1}}(\sigma ){{K}_{1}}(\sigma r),\,\,\,\,r > 1, \hfill \\ \end{gathered} \right.$$

(8)

where $\tilde {\sigma }$ is vortex intensity; ${{\sigma }} = \sqrt {{b \mathord{\left/ {\vphantom {b {{{U}_{0}}}}} \right. \kern-0em} {{{U}_{0}}}}}$, $b = {{{{\beta }}{{L}^{2}}} \mathord{\left/ {\vphantom {{{{\beta }}{{L}^{2}}} {{{U}_{0}}}}} \right. \kern-0em} {{{U}_{0}}}}$ is a planetary parameter; ${{\beta }}$ is Rossby parameter, which characterizes beta-effect; L is dimensional radius of the vortex; ${{U}_{0}}$ is characteristic flow velocity; ${{K}_{1}}$, ${{I}_{1}}$ are modified Bessel functions (Bessel functions of imaginary argument) of the first order.

For the vertical velocity $w\left( {r,z} \right)$, we have an expression from continuity equation in cylindrical coordinates:

$$\frac{{\partial w(r,z)}}{{\partial z}} = - \frac{{\partial {{{v}}_{r}}}}{{\partial r}} - \frac{{\partial {{{v}}_{\varphi }}}}{{r\partial \varphi }} - \frac{{{{{v}}_{r}}}}{r}.$$

(9)

Taking into account that in virtue of radial symmetry, ${{{{\delta }}{{{v}}_{\varphi }}} \mathord{\left/ {\vphantom {{{{\delta }}{{{v}}_{\varphi }}} {r{{\delta }}\varphi }}} \right. \kern-0em} {r{{\delta }}\varphi }} = 0$, we have from (10)

$$\frac{{\partial w(r,z)}}{{\partial z}} = - \frac{{\partial {{{v}}_{r}}}}{{\partial r}} - \frac{{{{{v}}_{r}}}}{r}.$$

(10)

Substituting ${{{v}}_{r}}\left( {r,z} \right)$ from (8) into (11) and integrating by z from the bottom boundary of ice z = 0 to horizon z, taking into account that the vertical velocity at this boundary is zero, we have

$$\begin{gathered} w(r,z) = - \sqrt {\frac{{{{E}_{{v}}}}}{{2f}}\left( {\frac{{d{v}_{\varphi }^{{({\text{g}})}}}}{{dr}} + \frac{{{v}_{\varphi }^{{({\text{g}})}}}}{r}} \right)} \hfill \\ \times \,\,\left[ {1 - \sqrt 2 \exp ( - \alpha z)\sin \left( {\alpha z + \frac{\pi }{4}} \right)} \right]. \hfill \\ \end{gathered} $$

(11)

Taking into account (9), we obtain

$$\begin{gathered} \frac{{d{v}_{\varphi }^{{({\text{g}})}}}}{{dr}} + \frac{{{v}_{\varphi }^{{({\text{g}})}}}}{r} \\ = - \tilde {\sigma }\left\{ \begin{gathered} {{K}_{1}}(\sigma )\left[ {\sigma \frac{{{{I}_{0}}(\sigma r) + {{I}_{2}}(\sigma r)}}{2} + \frac{{{{I}_{1}}(\sigma r)}}{r}} \right],\,\,\,\,r \leqslant 1 \hfill \\ {{I}_{1}}(\sigma )\left[ { - \sigma \frac{{{{K}_{0}}(\sigma r) + {{K}_{2}}(\sigma r)}}{2} + \frac{{{{K}_{1}}(\sigma r)}}{r}} \right],\,\,\,\,r > 1. \hfill \\ \end{gathered} \right. \\ \end{gathered} $$

(12)

${{K}_{0}}$, ${{I}_{0}}$ and ${{K}_{2}}$, ${{I}_{2}}$ are modified Bessel functions of zero and second order, respectively.

The obtained formulas for ${{{v}}_{r}}\left( {r,z} \right)$, ${{{v}}_{\varphi }}\left( {r,z} \right)$, and $w\left( {r,z} \right)$ allow us to calculate the three-dimensional structure of the flow in the under-ice Ekman boundary layer, generated by geostrophic canyon vortex. Figure 6 gives the calculated velocity field ${{{v}}_{r}}\left( {r,z} \right)$, $w\left( {r,z} \right)$ in the plane of variables $\left( {r,z} \right)$.

Fig. 6.

(a) Vector field of dimensionless velocities (${{{v}}_{r}}\left( {r,z} \right)$, $w\left( {r,z} \right)$) in the vertical plane; r = 1―the lateral boundary of the geostrophic vortex; dashed lines are boundaries of the Stewartson layer with a thickness of $E_{h}^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0em} 4}}}$ (${{E}_{h}} = {{{{A}_{L}}} \mathord{\left/ {\vphantom {{{{A}_{L}}} {\left( {{{f}_{0}}{{L}^{2}}} \right)}}} \right. \kern-0em} {\left( {{{f}_{0}}{{L}^{2}}} \right)}}$ is the horizontal Ekman number) and under-ice Ekman layer with a thickness of $E_{{v}}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}$; (b) more detail velocity field (${{{v}}_{r}}\left( {r,z} \right)$, $w\left( {r,z} \right)$) in the vertical domain in the zone of crossing of Stewartson layer with under-ice Ekman layer.

Figure 6 shows that, in the inner part of Stewartson layer, warmer deep water rises into the cold surface Ekman layer, which is in contact with ice. The horizontal and vertical water motions are most active in the Ekman layer in the zone of crossing with Stewartson layer (Fig. 5a).

As can be seen from Figs. 6, 7, a divergent vortex forms under ice in the ring domain of crossing the Stewartson and Ekman layers in the form of a horizontal vortex torus, and the melting of ice from below in the form of a ring is precisely due to the formation of this vortex torus.

Fig. 7.

Radial distribution of the calculated horizontal flow velocities (${{{v}}_{r}}\left( {r,z} \right)$, ${{{v}}_{\varphi }}\left( {r,z} \right)$) from the center of ring in the Ekman layer under ice at $z = 20$ m. The stroke shows the tangent to the contour of the geostrophic vortex (the center of the ring). The maximal length of the arrow corresponds to the speed of 7–10 cm/s.

DISCUSSION OF RESULTS

As follows from the presented theory, the ring structure in ice is due to the formation of a divergent vortex in under-ice Ekman layer. In this layer, Ekman inflow is generated, which in this case is directed upward. Note that in open water, Ekman inflow depends on wind vorticity and can be directed either upward or downward.

Ice melting from below in the ring domain is due to the formation of a vertical circular Stewartson layer on the lateral surface of the geostrophic vortex and, as a consequence, the formation of a vortex torus in the zone of crossing of cylindrical Stewartson layer and the under-ice Ekman layer. Warm deep water rises through the inner part of the Stewartson layer into the Ekman layer and further into the outer domain. The most intense horizontal and vertical motions of water take place in the zone of intersection of these layers.

Note that the vertical velocities in the topographic vortices in a homogeneous fluid are small; therefore, they are neglected, and the motions are assumed quasi-two-dimensional. However, Ekman intake in the under-ice Ekman layer initiates considerable vertical motions in the Stewartson layer. From this it follows that lens-like structures under rings are most likely not an independent formation, but form under the effect of a divergent vortex in the under-ice Ekman layer.

The hydrological studies of water structure in sections through ring structures in 2012, 2014, and, especially, in February–March 2019, showed that lens-like vortices under the rings have an isolated round shape and radial symmetry. The anomalous water structure observed in 2019 in areas of the ice ring at Nizhnee Izgolov’e Cape and the measurements of under-ice currents indicate that the ice rings are directly related to the anticyclonic vortex under ice and the ice ring is a surface manifestation of under-ice vortex tori.

Note that the field studies of winter 2019 near Nizhnee Izgolov’e Cape confirmed the lens-like structure of water under ice ring for the temperature field; this structure was first detected by N.G. Granin et al. in 2009 [3]. The temperature in vortex core was 1–1.25°C. In addition, Granin et al. [3] found less mineralized water under ice in the center of the ring compared with its marginal area in the layer 0–40 m; however, interestingly, no increase in mineralization was observed below. An assumption was made that the anomalous hydrochemical structure of water is a result of rise of deep water with low salinity and higher temperature through the thermocline, and this rise can be due to the presence of gas hydrates in bottom sediments; i.e., the authors of this article adhere to the gas hypothesis of the formation of ice rings.

CONCLUSIONS

The study [13] proposed two hypotheses of vortex formation. The first is that the vortices form under ice when the ice cover is stable. In winter, under synoptic variations of atmospheric pressure, the mechanical energy is constantly transferred through ice from the atmosphere into water mass. As well as in the ice-free period, seiche, internal waves, and setups form, their amplitude being comparable with those in summer and autumn [6]. The second hypothesis implies that the vortices form in some moment between the periods of autumn vertical mixing and ice formation. After vertical mixing, warmer (and lighter) water stays near the surface, while the vortex sinks to the depth of neutral buoyancy level. Surface vortices and ice rings are rare in early winter with only one exception observed near Nizhnee Izgolov’e Cape in 2009. In winter and, especially, spring, when the top water layer starts warming up and becomes lighter, the vortex rises to reach the ice cover.

In study [14], the same authors state that the vortex under ice forms long before ice formation as the result of wind field effect and water export from Barguzin Bay around Nizhnee Izgolov’e Cape.

This study proposes another theory―the geostrophic vortices under ice in the zone of ice ring formation are due to bed relief, namely, canyons, and the generation of a new types of topographic vortices―canyon vortices. All other effects, including lens-type formations in the field of temperature under ice, are manifestations of the evolution of these canyon vortices.